Open Access Article

This Open Access Article is licensed under a

Creative Commons Attribution 3.0 Unported Licence

DOI: 10.1039/C8SC00529J
(Edge Article)
Chem. Sci., 2018, Advance Article

Ivan Duchemin*^{a},
Ciro A. Guido^{bc},
Denis Jacquemin^{b} and
Xavier Blase*^{d}
^{a}Univ. Grenobles Alpes, CEA, INAC-MEM, L_Sim, F-38000 Grenoble, France. E-mail: ivan.duchemin@cea.fr; xavier.blase@neel.cnrs.fr
^{b}Laboratoire CEISAM – UMR CNR 6230, Université de Nantes, 2 Rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
^{c}Laboratoire MOLTECH – UMR CNRS 6200, Université de Angers, 2 Bd Lavoisier, 49045 Angers Cedex, France
^{d}Univ. Grenobles Alpes, CNRS, Institut Néel, F-38042 Grenoble, France

Received
1st February 2018
, Accepted 2nd April 2018

First published on 5th April 2018

The Bethe–Salpeter equation (BSE) formalism has been recently shown to be a valuable alternative to time-dependent density functional theory (TD-DFT) with the same computing time scaling with system size. In particular, problematic transitions for TD-DFT such as charge-transfer, Rydberg and cyanine-like excitations were shown to be accurately described with BSE. We demonstrate here that combining the BSE formalism with the polarisable continuum model (PCM) allows us to include simultaneously linear-response and state-specific contributions to solvatochromism. This is confirmed by exploring transitions of various natures (local, charge-transfer, etc.) in a series of solvated molecules (acrolein, indigo, p-nitro-aniline, donor–acceptor complexes, etc.) for which we compare BSE solvatochromic shifts to those obtained by linear-response and state-specific TD-DFT implementations. Such a remarkable and unique feature is particularly valuable for the study of solvent effects on excitations presenting a hybrid localised/charge-transfer character.

As another alternative to TD-DFT, the Bethe–Salpeter equation (BSE) formalism^{31–35} has been recently experiencing a growing interest in the study of molecular systems due to its ability to overcome some of the problems that TD-DFT is facing, including charge-transfer^{36–44} and cyanine-like^{45,46} excitations, while preserving the same scaling in its standard implementations. Extensive benchmark studies on diverse molecular families have been performed,^{47–53} demonstrating that excellent agreement with higher-level many-body wavefunction techniques, such as coupled-cluster (CC3) or CASPT2, could be obtained for all types of transitions, provided that they do not present a strong multiple-excitation character. Singlet–triplet transitions constitute the only notable exception as they may present the same instability problems with BSE and TD-DFT.^{51,52} We have recently reviewed the differences between BSE and TD-DFT formalisms in a chemical context, and we refer the interested reader to that original work for more details.^{35}

As compared to TD-DFT, the BSE formalism relies on transition matrix elements between occupied and virtual energy levels calculated at the GW level, where G and W stand for the one-body Green's function and the screened-Coulomb potential. These GW energy levels, including HOMO and LUMO frontier orbital energies, were shown to be in much better agreement with reference wavefunction calculations as compared to standard DFT Kohn–Sham (KS) eigenvalues. Namely, GW HOMO and LUMO energies can be directly associated with the ionisation potential (IP) and electronic affinity (AE). Further, the TD-DFT exchange-correlation kernel matrix elements in the occupied-to-virtual transition space are replaced within the BSE formalism by matrix elements involving the screened Coulomb potential interaction between the hole and the electron.

The coupling of the GW and PCM formalisms was recently achieved,^{54} within an integral equation formalism^{9} (IEF-PCM) implementation including the so-called non-equilibrium (neq) effects related to the separation of the solvent response into its “fast” electronic and “slow” nuclear contributions. The electron and hole polarisation energies, namely the shifts of the ionisation potential and electronic affinity from the gas phase to solution, were very well reproduced with GW taking as a reference standard ΔSCF + PCM calculations where total energy calculations of the solvated ions and neutral species were performed at the DFT or CCSD levels. Similar studies were also performed adopting discrete polarisable models to study organic semiconductors and complexes of interest for optoelectronic applications.^{55–58} As central (GW + PCM) formalism features, we underline that the polarisation energies for all occupied (P_{i}^{+}) and virtual (P_{a}^{−}) energy levels can be calculated, not only those associated with frontier orbitals, and that these polarisation energies are SS. Such polarisation energies, of the order of an eV in the case of water solvated nucleobases,^{54} dramatically reduce the HOMO–LUMO gap as compared to its gas phase value (see Fig. 1).

In the present study, we demonstrate that the BSE formalism combined with the PCM in a non-equilibrium formulation intrinsically combines the LR and SS solvatochromic contributions associated with the effect of the polarisable environment. Such a remarkable feature hinges in particular on the proper inclusion of dynamic polarisation energies not only for the occupied and virtual energy levels, calculated within the GW formalism, but also for the screened Coulomb electron–hole interaction. This renormalisation by polarisation of energy levels and electron–hole interactions leads to the SS contribution to solvatochromic shifts, in addition to the familiar transition matrix elements stemming from the LR contributions. This simultaneous account of both LR and SS effects was, to the best of our knowledge, only achieved using an a posteriori sum of the two terms in the context of the computationally more expensive ADC(2)/COSMO approach,^{26} and more recently with a TD-DFT/discrete polarizable scheme.^{20} Here, the obtained (BSE + PCM) solvatochromic shifts are computed for paradigmatic transitions in acrolein, indigo, p-nitro-aniline (PNA), a small donor–acceptor complex and a solvatochromic probe (see Fig. 2), and are compared to the sum of the shifts obtained at the TD-DFT level within the standard LR and SS implementations of the PCM, respectively.

(1) |

(2) |

(3) |

While eqn (3) is exact, an expression for Σ^{XC} should be defined. The GW formalism provides an approximation for Σ^{XC} to first order in the screened Coulomb potential W, with:

(4) |

(5) |

(6) |

(7) |

The GW eigenvalues have been shown in many benchmark studies on gas phase molecular systems to be significantly more accurate than KS or Hartree–Fock eigenstates, providing, e.g., frontier orbital energies within a few tenths of an eV with respect to reference CCSD(T) calculations.^{68–72} For the sake of illustration, our gas phase ionization potential (IP) and electronic affinity (EA) are 10.35 eV (−ε^{GW}_{HOMO}) and 0.68 eV (+ε^{GW}_{LUMO}) respectively for acrolein within our GW approach, comparable to 10.01 eV and 0.70 eV within CCSD(T) for the same atomic basis (cc-pVTZ) and the same geometry (see the ESI†). In practice, GW calculations proceed traditionally by correcting input KS eigenstates, substituting the GW self-energy contribution to the DFT exchange-correlation potential:

(8) |

The obtained ε^{GW}_{a/i} eigenvalues and the screened Coulomb potential W serve as input quantities for the BSE excitation energy calculation.

(9) |

A^{BSE}_{ai,bj} = δ_{ab}δ_{ij}(ε^{GW}_{a} − ε^{GW}_{i}) − 〈ab|W|ij〉 + 〈ai|bj〉
| (10) |

〈ab|W|ij〉 = 〈ϕ_{a}(r)ϕ_{b}(r)W(r,r′)ϕ_{i}(r′)ϕ_{j}(r′)〉
| (11) |

〈ai|bj〉 = 〈ϕ_{a}(r)ϕ_{i}(r)v(r,r′)ϕ_{b}(r′)ϕ_{j}(r′)〉
| (12) |

(13) |

W = ṽ + ṽχ^{QM}_{0}W
| (14) |

ṽ = v + vχ^{PCM}_{0}ṽ
| (15) |

ṽ = v + vχ^{PCM}v
| (16) |

Importantly, in the present non-equilibrium formulation of the (BSE/GW + PCM) implementation, the reaction field that modifies W is associated with the “fast” electronic excitations, namely with the ε_{∞} dielectric constant (e.g., ε_{∞} = 1.78 in water) that is the low-frequency optical dielectric response. The inclusion of the “slow” response of the solute (given by, e.g., ε_{0} = 78.35 for water) is accounted for in the preliminary ground-state DFT + PCM(ε_{0}) run that serves as a starting point for GW and BSE calculations, namely in the construction of χ_{0} and G in eqn (5) and (7). The flow of calculations can then be summarised as follows:

(1) Calculation of input KS {ε_{n}, ϕ_{n}} eigenstates within a DFT + PCM(ε_{0}) scheme. These eigenstates contain ground-state solvation effects,

(2) Calculation of the “fast” reaction field v^{reac}(ε_{∞}) in an auxiliary basis representation (see ref. 54, 56 and 58) that is incorporated inside the screened Coulomb potential W following eqn (5) and (7),

(3) Correction of the {ε_{n}} KS eigenstates, that include “slow” polarisation effects, with the GW self-energy operator that contains the “fast” v^{reac}(ε_{∞}) reaction field in order to yield the GW + PCM neq {ε_{n}^{GW+PCM}} energy levels,

(4) Resolution of the BSE excitation energy eigenvalue problem with the {ε_{n}^{GW+PCM}} eigenstates and the W and ṽ potentials that include v^{reac}(ε_{∞}) = vχ^{PCM}(ε_{∞})v.

(17a) |

+δ_{ab}δ_{ij}Δ(ε^{GW/PCM}_{a} − ε^{GW/PCM}_{i}) − 〈ab|ΔW|ij〉
| (17b) |

+〈ai|vχ^{PCM}(ε_{∞})v|bj〉,
| (17c) |

As the easiest identification, the 3^{rd} line contribution (eqn (17c)) straightforwardly corresponds to LR reaction field matrix elements, i.e., to transition density polarisation effects. More explicitly, 〈ai|vχ^{PCM}v|bj〉 describes the action on ϕ_{b}(r′)ϕ_{j}(r′) of the reaction field , where is the surface charge generated by the PCM susceptibility χ^{PCM}(ε_{∞}) in response to the field generated by the transition density ϕ_{a}(r)ϕ_{i}(r). In the original notations of ref. 10, such matrix elements are strictly equivalent to the B^{f}_{ai,bj} = 〈ai|K^{f}|bj〉 linear response terms (eqn (31) in ref. 10) where K^{f} is the “fast” reaction potential integral operator. As such, the BSE + PCM formalism includes the LR solvent contributions.

Let us now demonstrate that the second line (eqn (17b)) recovers the SS contribution. Part of the demonstration relies on the specificity of the GW + PCM formalism that captures accurately SS dynamic polarisation energies, namely:

(18a) |

(18b) |

(19) |

In the case of a “pure” transition between levels (i) and (a), one straightforwardly obtains:

(20) |

(21) |

We now turn to the case of general BSE electron–hole eigenstates assuming for simplicity the TDA approximation. Hole and electron densities are now described by a correlated codensity ρ_{λ}(r_{e},r_{h}) = |ψ_{λ}(r_{e},r_{h})|^{2} that cannot be expressed in terms of individual eigenstate densities as in the simple two-level model. This however does not affect the central result that the screened Coulomb potential W, and thus the electron–hole interaction:

(22) |

Concerning now the contribution from the GW electron and hole quasiparticle energies, we obtain, considering e.g. the unoccupied (electron) energy levels ε^{GW/PCM}_{a}:

(23) |

Within the present neq approach, the starting DFT calculations are performed in combination with the COSMO formalism^{82} as implemented in NWChem, using the (ε_{0}) dielectric constant associated with the (slow) nuclear and electronic degrees of freedom (e.g., ε_{0} = 78.39 for water). As such, the KS eigenstates used to build our electronic excitations carry information regarding the ground-state polarisation effects. In a second-step, the screened Coulomb potential used in the GW and BSE calculations, that describes fast electronic excitations out of the ground-state, is “dressed” with the reaction field generated with the (fast) electronic dielectric response in the low-frequency limit (e.g., ε_{∞} = 1.78 for water). In GW and BSE, the reaction field is described at the full IEF-PCM level, following the implementation detailed in ref. 54.

Our BSE calculations are compared to TD-DFT calculations performed with the Gaussian16 package^{83} using the IEF-PCM non-equilibrium implementation and the same cc-pVTZ atomic basis set. The LR shifts have been obtained with the default Gaussian16 implementation,^{10,11} whereas the SS shifts have been determined with the so-called corrected LR (cLR) formalism,^{13} that is a perturbative approach. For the TD-DFT calculations we selected the same PBE0 functional, complemented in the case of CT transitions with calculations performed with the range-separated hybrid CAM-B3LYP functional,^{84} known to more accurately describe CT transitions in the TD-DFT context. The character of the electronic transitions in TD-DFT has been determined by estimating the effective electronic displacement induced by the excitation using the Γ index.^{85,86}

In order to differentiate static and dynamic contributions within our approach, we make use of a ground state frozen polarization excitation energy (Ω_{0}),^{15,19} obtained by considering that the polarisable medium does not respond to the fast electronic excitation, namely by setting ε_{∞} = 1 while keeping the correct static dielectric constant of the solvent ε_{0}. As such, labelling Ω the final BSE excitation energies, accounting for both static and dynamic PCM responses, the quantity (Ω − Ω_{0}) quantifies the impact of switching on the fast PCM(ε_{∞}) response. We underline that our BSE calculations are performed beyond the Tamm–Dancoff approximation (TDA), that is, include the full BSE matrix. However, the inclusion of contributions from de-excitation processes complicates the simple analysis provided above concerning the LR and SS contributions with a clear distinction between 〈ab|W|ij〉 terms, namely the coupling between density terms, and 〈ai|v^{reac}|bj〉 contributions, that is the coupling between transition dipoles. Beyond TDA, we can decompose the expectation value 〈Ψ_{BSE}|H_{BSE}|Ψ_{BSE}〉 into

(24a) |

(24b) |

(24c) |

Our data are compiled in Table 1. The gas phase (Ω_{g}) and solvated (Ω) theoretical and experimental excitation energies are provided, together with the overall shift (Ω − Ω_{g}). In our non-equilibrium formalism, the shift is decomposed into a ground-state static contribution (Ω_{0} − Ω_{g}) and a dynamic one (Ω − Ω_{0}) that is itself partitioned into LR and SS contributions. Within TD-DFT, the LR and SS (cLR) shifts are obtained as two separate calculations, whereas within BSE, this decomposition is obtained by partitioning (see above) the non-equilibrium BSE/GW + PCM shift. Wavefunction approaches, such as ADC(2) or CCSD, can also be used to provide both contributions simultaneously^{26} or separately^{25} and we give some literature examples in Table 1.

Ω_{g} |
Ω(Ω − Ω_{g}) |
Ω_{0} − Ω_{g} |
Ω − Ω_{0} |
Ref. | ||
---|---|---|---|---|---|---|

LR | SS | |||||

a In the breakdown approach used in that work, the SS contribution is +0.17 eV and the LR contribution is negligible.b In the breakdown approach used in that work, the SS contribution is −0.21 eV and the LR contribution is −0.18 eV.c In ethanol, the most polar protic solvent in which indigo is soluble experimentally. | ||||||

Acrolein n–π* in water | ||||||

TD-PBE0 (LR) | 3.599 | 3.785(+0.186) | +0.189 | −0.003 | This work | |

TD-PBE0 (cLR) | 3.599 | 3.736(+0.137) | +0.189 | −0.052 | This work | |

BSE | 3.736 | 3.988(+0.252) | +0.232 | −0.011 | +0.031 | This work |

CC3 | 3.74 | This work | ||||

CCSDR(3)/MM | 3.81 | 4.08(+0.27) | 91 | |||

SAC-CI | 3.85 | 3.95(+0.10) | 21 | |||

CCSD | 3.94 | 4.14(+0.20) | 22 | |||

ADC(2) | 3.69 | 3.86(+0.17)^{a} |
26 | |||

CCSD (LR) | 3.88 | 4.10(+0.22) | 25 | |||

CCSD (SS) | 3.88 | 4.05(+0.17) | 25 | |||

Exp. | 3.69 | 3.94(+0.25) | 87 and 88 | |||

Acrolein π–π* in water | ||||||

TD-PBE0 (LR) | 6.383 | 6.174(−0.209) | −0.073 | −0.136 | This work | |

TD-PBE0 (cLR) | 6.383 | 6.281(−0.102) | −0.073 | −0.029 | This work | |

BSE | 6.498 | 6.214(−0.284) | −0.112 | −0.163 | −0.004 | This work |

CC3 | 6.82 | This work | ||||

CCSDR(3)/MM | 6.73 | 6.22(−0.51) | 91 | |||

SAC-CI | 6.97 | 6.75(−0.22) | 21 | |||

CCSD | 6.89 | 6.54(−0.35) | 22 | |||

ADC(2) | 6.79 | 6.40(−0.39)^{b} |
26 | |||

CCSD (LR) | 6.80 | 6.39(−0.41) | 25 | |||

CCSD (SS) | 6.80 | 6.54(−0.26) | 25 | |||

Exp. | 6.42 | 5.89(−0.53)^{a} |
87 and 88 | |||

Indigo in water | ||||||

TD-PBE0 (LR) | 2.304 | 2.160(−0.144) | −0.068 | −0.076 | This work | |

TD-PBE0 (cLR) | 2.304 | 2.229(−0.075) | −0.068 | −0.007 | This work | |

BSE | 2.259 | 2.047(−0.212) | −0.122 | −0.082 | −0.008 | This work |

Exp. | 2.32 | 2.04(−0.19)^{c} |
92, 93, 94, 95, 96, 97 and 98 |

The acrolein n–π* transition in the gas phase (Ω_{g}) is found to be located at 3.60 eV and 3.74 eV within TD-PBE0 and BSE respectively, in good agreement with the CC3 value of 3.74 eV, as well as with previous wavefunction estimates and experiment. The analysis of the intermediate (Ω_{0} − Ω_{g}) and total (Ω − Ω_{g}) solvatochromic shift indicates that the positive solvatochromism for this transition is entirely dominated by ground-state effects and that the additional shift associated with the fast optical excitation is negligible. TD-PBE0 and BSE calculations performed on top of the DFT + PCM(ε_{0}) ground-state yield similar Ω_{0} − Ω_{g} shifts, namely +0.19 eV and +0.23 eV, respectively. Concerning the effect of switching ε_{∞} (1.78 in water), both TD-PBE0+PCM, within LR or cLR, and BSE + PCM yield very small additional shifts, ranging from −0.05 eV (cLR) to +0.02 eV (BSE). The BSE computed shift of +0.25 eV turns out to be in close agreement with the experimental values (+0.25 eV), as well as with previous wavefunction approaches (see Table 1).

While the n–π* transition does not allow us to clearly discriminate between LR and SS responses, a more interesting test of the effect of the fast response (ε_{∞}) polarisable environment comes with the higher-lying π–π* transition. For this transition, the reaction field associated with the optical excitation, namely v^{reac}(ε_{∞}), leads to a shift that is larger than the one associated with ground-state solvation charges. In the gas phase, the BSE transition energy (6.50 eV) is reasonably close to the CC3 (6.82 eV) and experimental (6.42 eV) values. The most salient feature is that within TD-DFT, only the LR scheme can significantly contribute to the redshift, while the cLR approach fails to deliver any sizeable solvation effect, with (Ω − Ω_{0}) being equal to −0.14 eV for the former model, and −0.03 eV for the latter. This effect was expected for a local π–π* transition not involving a strong density reorganisation between the two states: the LR-PCM-TD-DFT is more suited as it captures the dominating contributions originating from the transition densities, that can be viewed as “dispersion-like” terms.^{19} At the CCSD level, the results of Caricato^{25} also demonstrated that the LR contribution is dominant; in that work the decomposition of the total response into various contributions was not performed, but rather two different models have been applied as in TD-DFT. We also underline that in their ADC(2) study, Lunkenheimer and Köhn also found that the LR term, negligible for the n–π* case,^{26} becomes large for the π–π* transition, though the approach used to compute the relative contributions is not straightforwardly comparable to ours. In any case, the BSE + PCM formalism, with a −0.17 eV (Ω − Ω_{0}) shift, captures the correct bathochromic effect. Further, consistent with the TD-DFT calculations, we observe that the BSE shift is dominated by the LR contribution, while the SS term provides a negligible shift. This shows, consistent with our analysis of eqn (17c), that the BSE formalism correctly captures the LR response. When compared to experiment, the BSE + PCM shift remains too small, but we recall that we neglect here, as in any continuum approach, the explicit solvent–solute interactions that are known to be significant in the present case.^{26,91}

To confirm the present observations, we consider the case of the lowest transition in indigo, a hallmark centro-symmetric dye presenting a low-lying dipole-allowed π–π* transition. This compound was studied previously at the TD-DFT level with the LR PCM model, and it was shown that this approach nicely reproduces the experimental solvatochromic shifts.^{99} With TD-DFT and the LR formalism, we found that the ground-state and dynamic polarisation effects, as measured respectively by (Ω_{0} − Ω_{g}) and (Ω − Ω_{0}), have the same order of magnitude, whereas the cLR correction does not lead to any significant (Ω − Ω_{0}) effect, as expected for a dye in which both the ground-state and excited-state total dipoles are strictly null. As in the case of the π–π* transition in acrolein, the BSE + PCM scheme leads to a clear redshift, with in this case a good agreement with the experimental trends as well. As a matter of fact, the (Ω − Ω_{0}) BSE LR (SS) shift is very close to the corresponding TD-DFT LR (SS) shift, demonstrating the relevance of the analysis and partitioning of the BSE overall (Ω − Ω_{0}) difference discussed in Section 3.

Ω_{g} |
Ω(Ω − Ω_{g}) |
Ω_{0} − Ω_{g} |
Ω − Ω_{0} |
||
---|---|---|---|---|---|

LR | SS | ||||

PNA_{perp} in water (“dark” CT excitation) |
|||||

TD-PBE0 (LR) | 3.686 | 3.316(−0.370) | −0.370 | 0.000 | |

TD-PBE0 (cLR) | 3.686 | 2.861(−0.795) | −0.370 | −0.425 | |

TD-CAM-B3LYP (LR) | 4.621 | 4.308(−0.313) | −0.312 | −0.001 | |

TD-CAM-B3LYP (cLR) | 4.621 | 4.038(−0.583) | −0.312 | −0.271 | |

BSE | 5.112 | 4.399(−0.713) | −0.423 | +0.015 | −0.304 |

Benzene–TCNE in water (“bright” CT excitation) | |||||

TD-PBE0 (LR) | 2.157 | 2.081(−0.076) | −0.065 | −0.011 | |

TD-PBE0 (cLR) | 2.157 | 1.747(−0.410) | −0.065 | −0.345 | |

TD-CAM-B3LYP (LR) | 2.944 | 2.876(−0.068) | −0.061 | −0.007 | |

TD-CAM-B3LYP (cLR) | 2.944 | 2.492(−0.452) | −0.061 | −0.291 | |

BSE | 3.503 | 3.121(−0.382) | −0.086 | −0.009 | −0.287 |

Exp. | 3.59 |

While the contribution of the fast optical dielectric response (ε_{∞}) to the solvatochromic shifts, as measured by (Ω − Ω_{0}), mainly originates from the LR contribution in both acrolein and indigo, the solvent-induced dynamic shift associated with CT transitions can only be described by adopting a SS (cLR here) formalism in the TD-DFT context. This is clearly illustrated by the PNA_{perp} system where the TD-DFT LR shift is trifling, a logical consequence of the dark nature of the considered transition, whereas the SS contribution is very large, as a result of the large density reorganisation associated with excitation. Consistent with previous studies,^{17} the magnitude of the TD-DFT SS shift strongly depends on the chosen functional, and it goes from −0.42 eV to −0.27 eV upon replacing the PBE0 functional by CAM-B3LYP, which is more suited for such an excited-state. The BSE (Ω − Ω_{0}) shift originates mainly from its SS component as well and lies in between the PBE0 and CAM-B3LYP values, though much closer to the latter, as expected.

The same conclusions are reached when considering the inter-molecular CT transition in the benzene–TCNE complex. First, we observe that the BSE formalism very nicely reproduces the gas phase excitation energy that is available experimentally. The TD-CAM-B3LYP calculation also provide a reasonable value, while the TD-PBE0 approach yields a much too small Ω_{g}, a logical consequence of its lack of long-range corrections. As expected for TD-DFT, the LR approach again provides a negligible (Ω − Ω_{0}) dynamic shift, while the SS formalism yields a large redshift. Again, the BSE + PCM formalism captures the correct physics, with a negligible LR contribution and a large SS contribution. Such calculation clearly demonstrates that the proposed BSE + PCM approach can also, following eqn (17b), capture SS dynamic shifts.

Ω_{g} |
Ω(Ω − Ω_{g}) |
Ω_{0} − Ω_{g} |
Ω − Ω_{0} |
||
---|---|---|---|---|---|

LR | SS | ||||

PNA in water (partial CT excitation) | |||||

TD-PBE0 (LR) | 4.202 | 3.802(−0.400) | −0.314 | −0.086 | |

TD-PBE0 (cLR) | 4.202 | 3.796(−0.406) | −0.314 | −0.092 | |

TD-CAM-B3LYP (LR) | 4.513 | 4.105(−0.408) | −0.321 | −0.087 | |

TD-CAM-B3LYP (cLR) | 4.513 | 4.005(−0.508) | −0.321 | −0.187 | |

BSE | 4.527 | 3.864(−0.663) | −0.470 | −0.090 | −0.102 |

4-Nitropyridine N-oxide in benzene (mixed excitation) | |||||

TD-PBE0 (LR) | 3.989 | 3.815(−0.174) | −0.048 | −0.126 | |

TD-PBE0 (cLR) | 3.989 | 3.797(−0.192) | −0.048 | −0.144 | |

TD-CAM-B3LYP (LR) | 4.196 | 4.023(−0.173) | −0.033 | −0.140 | |

TD-CAM-B3LYP (cLR) | 4.196 | 4.109(−0.087) | −0.033 | −0.054 | |

BSE | 3.966 | 3.658(−0.267) | −0.001 | −0.193 | −0.073 |

4-Nitropyridine N-oxide in water (mixed excitation) | |||||

TD-PBE0 (LR) | 3.989 | 3.797(−0.192) | −0.097 | −0.095 | |

TD-PBE0 (cLR) | 3.989 | 3.827(−0.162) | −0.097 | −0.065 | |

TD-CAM-B3LYP (LR) | 4.196 | 4.028(−0.168) | −0.065 | −0.103 | |

TD-CAM-B3LYP (cLR) | 4.196 | 4.067(−0.129) | −0.065 | −0.064 | |

BSE | 3.966 | 3.687(−0.279) | −0.070 | −0.141 | −0.068 |

4-Nitropyridine N-oxide is an organic probe used to assess the nature of solvents following a Kamel–Taft type of analysis.^{103} A previous throughout theoretical analysis of the solvatochromism of this probe is available,^{104} and demonstrates that, none of the available LR or SS PCM model is able to describe the solvent effects in a TD-DFT context, as the observed spectral changes come from a fine interplay of several effects. For this compound, the obtained conclusions are similar to the PNA case: both the LR and SS contributions are non-negligible even though the SS dynamic shift is somehow larger. Adding the LR and SS TD-DFT contributions, the overall (Ω − Ω_{0}) shift amounts to −0.16 eV and −0.17 eV with PBE0 and CAM-B3LYP in water, respectively, a shift that can be compared to the −0.21 eV value obtained within BSE. The experimental gas-phase excitation energy obtained through extrapolation in ref. 104 is 3.80 eV, and one notices that the BSE value is reasonably close to that estimate. In benzene, the measurement gives 3.52 eV,^{103} corresponding to a solvatochromic shift of −0.28 eV, a value that BSE can reproduce (−0.27 eV), whereas TD-DFT in unable to do so. In water, the hydrogen bonds with the negatively charged oxygen atom of the probe play a crucial role, and a blueshift is observed (the excitation maximum takes place at 3.94 eV),^{103} an effect that all continuum approaches logically fail to capture.

Fig. 3 Schematic representation of the (Ω − Ω_{0}) dynamic solvent induced shifts for the π–π* excitation in acrolein, the CT excitation in the benzene–TCNE complex, and the (planar) PNA mixed excitation. By dynamic shift, we mean the effect of switching the fast v^{reac}(ε_{∞}) PCM reaction field on top of the (slow) ground-state v^{reac}(ε_{0}) PCM response. The represented data correspond to the (Ω − Ω_{0}) shifts given in Tables 1–3. |

For the π–π* transition in acrolein, with a shift captured by the LR scheme only at the TD-DFT level, the decomposition of the (Ω − Ω_{0}) energy difference confirms that the variation of the BSE SS-like 〈Δε − W〉 component (eqn (17b)) does not contribute to the shift. Such a result may seem surprising since, as expected, the 〈Δε^{GW}〉 HOMO–LUMO gap becomes smaller by 2.34 eV upon “switching” v^{reac}(ε_{∞}), consistent with an (absolute) polarisation energy of ca. 1.1–1.2 eV that affects the IP and AE of hydrated compounds compared to the gas phase.^{54} However, very remarkably, the 〈W〉 electron–hole binding energy is also decreased by 2.35 eV. Namely, the screening by the fast reaction field reduces similarly the occupied-to-virtual (ε_{a}^{GW} − ε_{i}^{GW}) GW gap and the electron–hole binding energy (see Fig. 1). As such, only the 〈v^{reac}(ε_{∞})〉 LR terms (eqn (17c)) explain the solvatochromic effect, consistent with the TD-DFT results.

We now turn to the opposite situation of a CT excitation for which the solvatochromic shift can only be described by SS implementations, such as cLR, within TD-DFT. In the case of the benzene–TCNE complex, the LR 〈ia|v^{reac}(ε_{∞})|bj〉 contributing matrix elements become very small, as a logical consequence of the small overlap between the donor and acceptor wavefunctions. As such the only contribution to the (Ω − Ω_{0}) shift originates from the variation of the SS-like 〈Δε − W〉 term. The impact of v^{reac}(ε_{∞}) leads to the reduction of the 〈Δε〉 average gap by 1.60 eV, while the electron–hole binding energy 〈W〉 reduces by a smaller 1.31 eV amount, accounting for most of the (Ω − Ω_{0}) = −0.29 eV redshift reported in Table 2.

An important consequence of the present analysis, showing that, in BSE/GW theory, SS shifts result from the competition between the reduction of the GW occupied-to-virtual (ε_{a}^{GW} − ε_{i}^{GW}) energy gaps and the electron–hole 〈ab|W|ij〉 binding energies, is that both electron–electron and electron–hole interactions must be treated on the same footing, namely here through the screened Coulomb potential W. In particular, BSE + PCM calculations starting from KS eigenstates generated with exchange-correlation functionals optimally-tuned, so as to generate the correct gas phase HOMO–LUMO gap, might not deliver the correct SS contribution to optical excitation energy shifts.

Finally, while Table 2 indicates a large variability of the SS (cLR) TD-DFT (Ω − Ω_{0}) dynamic shift as a function of the chosen XC functional, it is interesting to emphasize the stability of the BSE/evGW data with respect to the input KS eigenstates used to build the evGW electronic energy levels and the screened Coulomb potential W. This is illustrated in Fig. 4a where we plot the PNA_{perp} excitation energy in water with the cLR-TD-PBE(α) (open triangles) and BSE/evGW@PBE(α) (open circles) methods. Here α indicates the percentage of exact exchange (EEX) used in the hybrid functional and going from 0% (PBE) to 100% (HFPBE) in Fig. 4. While the TD-DFT excitation energy is shown to increase steeply as a function of α, the BSE data are much more stable, with a variation of ca. 0.12 eV from (α = 0%) to (α = 90%). The stability of gas phase BSE/evGW excitation energies with respect to the starting KS starting point has been documented in previous benchmark studies,^{47,49} and is therefore shown here to pertain for condensed-phase calculations. In fact, plotting now in Fig. 4b the dynamic (Ω − Ω_{0}) solvatochromic shifts, we observe again a large variability of the TD-PBE(α) cLR shifts (the LR contribution is vanishingly small), while again the BSE shifts are extremely stable, evolving from −0.29 eV to −0.25 eV with α. To rationalize this result, we recall that BSE calculations are performed on top of partially self-consistent evGW calculations where the corrected electronic energy levels are self-consistently reinjected during the construction of G and W. This self-consistent treatment of the quasiparticle energies {ε_{n}^{GW}} and screened Coulomb potential W leads to a large stability of the BSE Hamiltonian and reaction field, with a small residual dependency on the starting KS eigenstates due to the fact that the {ϕ_{n}} eigenstates are unchanged. Such a stability of the BSE/evGW excitation energies and solvent induced shifts, that allows us to alleviate the standard problem of the proper choice of the XC functional central in TD-DFT, is one of the interesting features of the present scheme. As a fair tribute to TD-DFT, we observe however that the cLR shift matches closely that of BSE for a large exact exchange ratio. It is indeed known that large amounts of exact exchange are required in TD-DFT to obtain an accurate description of the ground-to-excited density variation for CT excitations.^{17}

Beyond the standard LR 〈ab|v^{reac}(ε_{∞})|ij〉 matrix element contributions, where v^{reac}(ε_{∞}) is the PCM reaction field to the electronic excitation, the proper inclusion of SS shifts hinges on the incorporation of v^{reac}(ε_{∞}) in the screened Coulomb potential W. This dressed screened Coulomb potential renormalises both electron–electron and electron–hole interactions, namely both the GW quasiparticle energies for occupied/virtual electronic levels and the BSE electron–hole (excitonic) binding energy. Ground-state polarisation effects are accounted for, in the present non-equilibrium scheme, by starting our BSE/GW calculations with KS eigenstates generated at the DFT + PCM(ε_{0}) level.

Following previous studies related to merging the GW formalism with discrete polarisable models,^{56–58} the same analysis can be straightforwardly applied to BSE calculations combined with other polarisation models.^{55,57,105} Specifically, the use of an explicit (molecular) description of the environment, combined with, e.g., empirical force fields, should allow us to account for ground-state electrostatic field effects induced by the solvent molecule static multipoles, an important contribution in the case of polar solvents that cannot be captured by the PCM model. As shown in the case of the π–π* transition in aqueous acrolein,^{89,91} such contributions can explain the difference between the experimental shift (−0.53 eV) and the ∼−0.25 eV redshifts obtained with the present BSE or TD-DFT-LR formalisms combined with the PCM.

ΔΣ^{GW} ≃ ΔΣ^{SEX} + ΔΣ^{COH},
| (25) |

(26) |

(27) |

(28) |

(29) |

with χ

- M. E. Casida, J. Mol. Struct.: THEOCHEM, 2009, 914, 3–18 CrossRef CAS.
- C. Ullrich, Time-Dependent Density-Functional Theory: Concepts and Applications, Oxford University Press, New York, 2012 Search PubMed.
- C. van Caillie and R. D. Amos, Chem. Phys. Lett., 1999, 308, 249–255 CrossRef CAS.
- F. Furche and R. Ahlrichs, J. Chem. Phys., 2002, 117, 7433–7447 CrossRef CAS.
- G. Scalmani, M. J. Frisch, B. Mennucci, J. Tomasi, R. Cammi and V. Barone, J. Chem. Phys., 2006, 124, 094107 CrossRef PubMed.
- J. Liu and W. Z. Liang, J. Chem. Phys., 2011, 135, 184111 CrossRef PubMed.
- C. A. Guido, G. Scalmani, B. Mennucci and D. Jacquemin, J. Chem. Phys., 2017, 146, 204106 CrossRef PubMed.
- S. Miertǔs, E. Scrocco and J. Tomasi, Chem. Phys., 1981, 55, 117–129 CrossRef.
- E. Cancès, B. Mennucci and J. Tomasi, J. Chem. Phys., 1997, 107, 3032–3041 CrossRef.
- R. Cammi and B. Mennucci, J. Chem. Phys., 1999, 110, 9877–9886 CrossRef CAS.
- M. Cossi and V. Barone, J. Chem. Phys., 2001, 115, 4708–4717 CrossRef CAS.
- R. Cammi, S. Corni, B. Mennucci and J. Tomasi, J. Chem. Phys., 2005, 122, 104513 CrossRef CAS PubMed.
- M. Caricato, B. Mennucci, J. Tomasi, F. Ingrosso, R. Cammi, S. Corni and G. Scalmani, J. Chem. Phys., 2006, 124, 124520 CrossRef PubMed.
- R. Improta, V. Barone, G. Scalmani and M. J. Frisch, J. Chem. Phys., 2006, 125, 054103 CrossRef PubMed.
- A. V. Marenich, C. J. Cramer, D. G. Truhlar, C. A. Guido, B. Mennucci, G. Scalmani and M. J. Frisch, Chem. Sci., 2011, 2, 2143–2161 RSC.
- A. Pedone, J. Chem. Theory Comput., 2013, 9, 4087–4096 CrossRef CAS PubMed.
- C. A. Guido, D. Jacquemin, C. Adamo and B. Mennucci, J. Chem. Theory Comput., 2015, 11, 5782–5790 CrossRef CAS PubMed.
- C. A. Guido, B. Mennucci, G. Scalmani and D. Jacquemin, J. Chem. Theory Comput., 2018, 14, 1544–1553 CrossRef CAS PubMed.
- S. Corni, R. Cammi, B. Mennucci and J. Tomasi, J. Chem. Phys., 2005, 123, 134512 CrossRef CAS PubMed.
- R. Guareschi, O. Valsson, C. Curutchet, B. Mennucci and C. Filippi, J. Phys. Chem. Lett., 2016, 7, 4547–4553 CrossRef CAS PubMed.
- R. Cammi, R. Fukuda, M. Ehara and H. Nakatsuji, J. Chem. Phys., 2010, 133, 024104 CrossRef PubMed.
- M. Caricato, B. Mennucci, G. Scalmani, G. W. Trucks and M. J. Frisch, J. Chem. Phys., 2010, 132, 084102 CrossRef PubMed.
- R. Fukuda, M. Ehara, H. Nakatsuji and R. Cammi, J. Chem. Phys., 2011, 134, 104109 CrossRef PubMed.
- M. Caricato, J. Chem. Theory Comput., 2012, 8, 4494–4502 CrossRef CAS PubMed.
- M. Caricato, J. Chem. Phys., 2013, 139, 044116 CrossRef PubMed.
- B. Lunkenheimer and A. Köhn, J. Chem. Theory Comput., 2013, 9, 977–994 CrossRef CAS PubMed.
- M. Caricato, Comput. Theor. Chem., 2014, 1040, 99–105 CrossRef.
- R. Fukuda and M. Ehara, J. Chem. Phys., 2014, 141, 154104 CrossRef PubMed.
- J.-M. Mewes, Z.-Q. You, M. Wormit, T. Kriesche, J. M. Herbert and A. Dreuw, J. Phys. Chem. A, 2015, 119, 5446–5464 CrossRef CAS PubMed.
- J.-M. Mewes, J. M. Herbert and A. Dreuw, Phys. Chem. Chem. Phys., 2017, 19, 1644–1654 RSC.
- E. E. Salpeter and H. A. Bethe, Phys. Rev., 1951, 84, 1232–1242 CrossRef.
- L. J. Sham and T. M. Rice, Phys. Rev., 1966, 144, 708–714 CrossRef CAS.
- W. Hanke and L. J. Sham, Phys. Rev. Lett., 1979, 43, 387–390 CrossRef CAS.
- G. Strinati, Phys. Rev. Lett., 1982, 49, 1519–1522 CrossRef CAS.
- X. Blase, I. Duchemin and D. Jacquemin, Chem. Soc. Rev., 2018, 47, 1022–1043 RSC.
- D. Rocca, D. Lu and G. Galli, J. Chem. Phys., 2010, 133, 164109 CrossRef PubMed.
- J. M. Garcia-Lastra and K. S. Thygesen, Phys. Rev. Lett., 2011, 106, 187402 CrossRef CAS PubMed.
- X. Blase and C. Attaccalite, Appl. Phys. Lett., 2011, 99, 171909 CrossRef.
- I. Duchemin, T. Deutsch and X. Blase, Phys. Rev. Lett., 2012, 109, 167801 CrossRef CAS PubMed.
- B. Baumeier, D. Andrienko and M. Rohlfing, J. Chem. Theory Comput., 2012, 8, 2790–2795 CrossRef CAS PubMed.
- S. Sharifzadeh, P. Darancet, L. Kronik and J. B. Neaton, J. Phys. Chem. Lett., 2013, 4, 2197–2201 CrossRef CAS.
- P. Cudazzo, M. Gatti, A. Rubio and F. Sottile, Phys. Rev. B, 2013, 88, 195152 CrossRef.
- V. Ziaei and T. Bredow, J. Chem. Phys., 2016, 145, 174305 CrossRef PubMed.
- D. Escudero, I. Duchemin, X. Blase and D. Jacquemin, J. Phys. Chem. Lett., 2017, 8, 936–940 CrossRef CAS PubMed.
- P. Boulanger, D. Jacquemin, I. Duchemin and X. Blase, J. Chem. Theory Comput., 2014, 10, 1212–1218 CrossRef CAS PubMed.
- C. Azarias, I. Duchemin, X. Blase and D. Jacquemin, J. Chem. Phys., 2017, 146, 034301 CrossRef PubMed.
- D. Jacquemin, I. Duchemin and X. Blase, J. Chem. Theory Comput., 2015, 11, 3290–3304 CrossRef CAS PubMed.
- F. Bruneval, S. M. Hamed and J. B. Neaton, J. Chem. Phys., 2015, 142, 244101 CrossRef PubMed.
- D. Jacquemin, I. Duchemin and X. Blase, J. Chem. Theory Comput., 2015, 11, 5340–5359 CrossRef CAS PubMed.
- D. Jacquemin, I. Duchemin, A. Blondel and X. Blase, J. Chem. Theory Comput., 2016, 12, 3969–3981 CrossRef CAS PubMed.
- D. Jacquemin, I. Duchemin, A. Blondel and X. Blase, J. Chem. Theory Comput., 2017, 13, 767–783 CrossRef CAS PubMed.
- T. Rangel, S. M. Hamed, F. Bruneval and J. B. Neaton, J. Chem. Phys., 2017, 146, 194108 CrossRef PubMed.
- D. Jacquemin, I. Duchemin and X. Blase, J. Phys. Chem. Lett., 2017, 8, 1524–1529 CrossRef CAS PubMed.
- I. Duchemin, D. Jacquemin and X. Blase, J. Chem. Phys., 2016, 144, 164106 CrossRef PubMed.
- B. Baumeier, M. Rohlfing and D. Andrienko, J. Chem. Theory Comput., 2014, 10, 3104–3110 CrossRef CAS PubMed.
- J. Li, G. D'Avino, I. Duchemin, D. Beljonne and X. Blase, J. Phys. Chem. Lett., 2016, 7, 2814–2820 CrossRef CAS PubMed.
- J. Li, G. D'Avino, A. Pershin, D. Jacquemin, I. Duchemin, D. Beljonne and X. Blase, Physical Review Materials, 2017, 1, 025602 CrossRef.
- J. Li, G. D'Avino, I. Duchemin, D. Beljonne and X. Blase, Phys. Rev. B, 2018, 97, 035108 CrossRef.
- M. Rohlfing and S. G. Louie, Phys. Rev. Lett., 1998, 80, 3320–3323 CrossRef CAS.
- L. X. Benedict, E. L. Shirley and R. B. Bohn, Phys. Rev. Lett., 1998, 80, 4514–4517 CrossRef CAS.
- S. Albrecht, L. Reining, R. Del Sole and G. Onida, Phys. Rev. Lett., 1998, 80, 4510–4513 CrossRef CAS.
- G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys., 2002, 74, 601–659 CrossRef CAS.
- G. Bussi, Phys. Scr., 2004, 2004, 141 CrossRef.
- L. Hedin, Phys. Rev., 1965, 139, A796–A823 CrossRef.
- M. S. Hybertsen and S. G. Louie, Phys. Rev. B, 1986, 34, 5390–5413 CrossRef CAS.
- R. W. Godby, M. Schlüter and L. J. Sham, Phys. Rev. B, 1988, 37, 10159–10175 CrossRef.
- B. Farid, R. Daling, D. Lenstra and W. van Haeringen, Phys. Rev. B, 1988, 38, 7530–7534 CrossRef.
- C. Faber, C. Attaccalite, V. Olevano, E. Runge and X. Blase, Phys. Rev. B, 2011, 83, 115123 CrossRef.
- K. Krause, M. E. Harding and W. Klopper, Mol. Phys., 2015, 113, 1952–1960 CrossRef CAS.
- F. Kaplan, M. E. Harding, C. Seiler, F. Weigend, F. Evers and M. J. van Setten, J. Chem. Theory Comput., 2016, 12, 2528–2541 CrossRef CAS PubMed.
- J. W. Knight, X. Wang, L. Gallandi, O. Dolgounitcheva, X. Ren, J. V. Ortiz, P. Rinke, T. Körzdörfer and N. Marom, J. Chem. Theory Comput., 2016, 12, 615–626 CrossRef CAS PubMed.
- T. Rangel, S. M. Hamed, F. Bruneval and J. B. Neaton, J. Chem. Theory Comput., 2016, 12, 2834–2842 CrossRef CAS PubMed.
- T. Stein, L. Kronik and R. Baer, J. Am. Chem. Soc., 2009, 131, 2818–2820 CrossRef CAS PubMed.
- I. Duchemin, J. Li and X. Blase, J. Chem. Theory Comput., 2017, 13, 1199–1208 CrossRef CAS PubMed.
- M. Valiev, E. Bylaska, N. Govind, K. Kowalski, T. Straatsma, H. V. Dam, D. Wang, J. Nieplocha, E. Apra, T. Windus and W. de Jong, Comput. Phys. Commun., 2010, 181, 1477–1489 CrossRef CAS.
- T. H. Dunning Jr, J. Chem. Phys., 1989, 90, 1007–1023 CrossRef.
- J. P. Perdew, M. Ernzerhof and K. Burke, J. Chem. Phys., 1996, 105, 9982–9985 CrossRef CAS.
- C. Adamo and V. Barone, J. Chem. Phys., 1999, 110, 6158–6170 CrossRef CAS.
- F. Weigend, J. Chem. Phys., 2002, 116, 3175–3183 CrossRef CAS.
- X. Blase, C. Attaccalite and V. Olevano, Phys. Rev. B, 2011, 83, 115103 CrossRef.
- C. Faber, J. L. Janssen, M. Côté, E. Runge and X. Blase, Phys. Rev. B, 2011, 84, 155104 CrossRef.
- A. Klamt, C. Moya and J. Palomar, J. Chem. Theory Comput., 2015, 11, 4220–4225 CrossRef CAS PubMed.
- M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery Jr, J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman and D. J. Fox, Gaussian-16 Revision A.03, Gaussian Inc., Wallingford CT, 2016 Search PubMed.
- T. Yanai, D. P. Tew and N. C. Handy, Chem. Phys. Lett., 2004, 393, 51–57 CrossRef CAS.
- C. A. Guido, P. Cortona and C. Adamo, J. Chem. Phys., 2014, 140, 104101 CrossRef PubMed.
- C. A. Guido, P. Cortona, B. Mennucci and C. Adamo, J. Chem. Theory Comput., 2013, 9, 3118–3126 CrossRef CAS PubMed.
- A. Moskvin, O. Yablonskii and L. Bondar, Theor. Exp. Chem., 1966, 2, 469 CrossRef.
- K. Aidas, A. Møgelhøj, E. J. K. Nilsson, M. S. Johnson, K. V. Mikkelsen, O. Christiansen, P. Söderhjelm and J. Kongsted, J. Chem. Phys., 2008, 128, 194503 CrossRef PubMed.
- J. M. Olsen, K. Aidas and J. Kongsted, J. Chem. Theory Comput., 2010, 6, 3721–3734 CrossRef CAS.
- A. H. Steindal, K. Ruud, L. Frediani, K. Aidas and J. Kongsted, J. Phys. Chem. B, 2011, 115, 3027–3037 CrossRef CAS PubMed.
- K. Sneskov, E. Matito, J. Kongsted and O. Christiansen, J. Chem. Theory Comput., 2010, 6, 839–850 CrossRef CAS PubMed.
- S. E. Sheppard and P. T. Newsome, J. Am. Chem. Soc., 1942, 64, 2937–3946 CrossRef CAS.
- P. W. Saddler, J. Org. Chem., 1956, 21, 316–318 CrossRef.
- M. Klessinger and W. Lüttke, Chem. Ber., 1966, 99, 2136–2145 CrossRef CAS.
- E. Wille and W. Lüttke, Angew. Chem., Int. Ed., 1971, 10, 803–804 CrossRef CAS.
- A. R. Monahan and J. E. Kuder, J. Org. Chem., 1972, 37, 4182–4184 CrossRef CAS.
- G. Haucke and G. Graness, Angew. Chem., Int. Ed., 1995, 34, 67–68 CrossRef CAS.
- R. Gerken, L. Fitjer, P. Müller, I. Usón, K. Kowski and P. Rademacher, Tetrahedron, 1999, 55, 14429–14434 CrossRef.
- D. Jacquemin, J. Preat, V. Wathelet and E. A. Perpète, J. Chem. Phys., 2006, 124, 074104 CrossRef PubMed.
- R. Cammi, L. Frediani, B. Mennucci and K. Ruud, J. Chem. Phys., 2003, 119, 5818–5827 CrossRef CAS.
- S. Millefiori, G. Favini, A. Millefiori and D. Grasso, Spectrochim. Acta, Part A, 1977, 33, 21–27 CrossRef.
- I. Hanazaki, J. Phys. Chem., 1972, 76, 1982–1989 CrossRef CAS.
- A. F. Lagalante, R. J. Jacobson and T. J. Bruno, J. Org. Chem., 1996, 61, 6404–6406 CrossRef CAS PubMed.
- S. Budzak, A. D. Laurent, C. Laurence, M. Miroslav and D. Jacquemin, J. Chem. Theory Comput., 2016, 12, 1919–1929 CrossRef CAS PubMed.
- D. Varsano, S. Caprasecca and E. Coccia, J. Phys.: Condens. Matter, 2017, 29, 013002 CrossRef PubMed.
- J. B. Neaton, M. S. Hybertsen and S. G. Louie, Phys. Rev. Lett., 2006, 97, 216405 CrossRef CAS PubMed.

## Footnote |

† Electronic supplementary information (ESI) available: Cartesian coordinates of the compounds. See DOI: 10.1039/c8sc00529j |

This journal is © The Royal Society of Chemistry 2018 |