The Bethe-Salpeter formalism with polarisable continuum embedding: reconciling linear-response and state-specific features.

Chemical Science View Article Online Open Access Article. Published on 05 April 2018. Downloaded on 17/05/2018 12:46:17. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. EDGE ARTICLE Cite this: Chem. Sci., 2018, 9, 4430 View Journal | View Issue The Bethe–Salpeter formalism with polarisable continuum embedding: reconciling linear-response and state-specific features† Ivan Duchemin, *a Ciro A. Guido, bc Denis Jacquemin b and Xavier Blase *d The Bethe–Salpeter equation (BSE) formalism has been recently shown to be a valuable alternative to timedependent 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 linearresponse and state-specific contributions to solvatochromism. This is confirmed by exploring transitions Received 1st February 2018 Accepted 2nd April 2018 of various natures (local, charge-transfer, etc.) in a series of solvated molecules (acrolein, indigo, p-nitroaniline, 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 DOI: 10.1039/c8sc00529j feature is particularly valuable for the study of solvent effects on excitations presenting a hybrid rsc.li/chemical-science localised/charge-transfer character. 1 Introduction The exploration of the excited-state (ES) properties of chemical systems certainly stands as a central question in theoretical chemistry. Indeed, ES phenomena govern many applications such as solar energy conversion, photocatalysis, light-emission or optical information storage. Further, while experimental characterisations can provide reference absorption and/or emission spectra, they are less suited to obtain some key information, e.g., ES geometries, nature of the excitation (localised, charge-transfer, etc.) or time evolution of hot electrons. Such a need for quantum mechanical formalisms allowing us to study realistic systems certainly explains the formidable popularity of time-dependent density functional theory (TD-DFT)1,2 that can be used to study the optical properties of systems comprising up to a few hundred atoms, thanks to a (formal) O ðN 4 Þ scaling with system size. Further, the availability of analytical TD-DFT derivatives3–7 together with the extension of efficient continuum models, such as the polarisable continuum model (PCM),8,9 to TD-DFT10–15 has a Univ. Grenobles Alpes, CEA, INAC-MEM, L_Sim, F-38000 Grenoble, France. E-mail: ; Laboratoire CEISAM – UMR CNR 6230, Université de Nantes, 2 Rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France b 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 † Electronic supplementary information (ESI) available: Cartesian coordinates of the compounds. See DOI: 10.1039/c8sc00529j 4430 | Chem. Sci., 2018, 9, 4430–4443 dramatically helped in bridging the gap between quantum simulations and realistic systems, by respectively allowing us to explore ES potential energy surfaces and to take into account the impact of the surroundings. In TD-DFT, the coupling with the PCM was initially performed within a linear-response (LR) formalism,10,11 that is, using the electronic transition density for including solvent effects. While such a LR model is generally accurate for describing the local ES, it is less suited for the charge-transfer (CT) ES, in which a large reorganisation of the electron density occurs. To tackle such an ES, state-specic (SS) PCM-TD-DFT models, in which the solvatochromic effects depend on the total electronic density of the ES, have been developed.7,13–15 At this stage, let us point out that, in using these SS models, one can encounter some cases for which the exact details of the chosen SS approach as well as the selected exchange-correlation functional have a very large impact on the results, especially when self-consistent iterative methods are selected.7,16–18 In addition, as rst pointed out by Corni et al.,19 who used a simple formal model explicitly including two states for the solute and two solvent macrostates, there is a need to simultaneously account for both LR and SS effects. However, to date, only the ad hoc sum of both LR and SS terms, determined in the context of a corrected linear response (cLR) approach,13 was proposed in a TD-DFT context.20 Alternatively one can turn towards single-reference electron-correlated wavefunction approaches, such as ADC(2), CC2, CCSD or SAC-CI, that have all been coupled to continuum models,21–30 but these models imply a signicantly increased computational effort compared to TDDFT. In this framework, we underline that the importance of the inclusion of both LR and SS effects was also clearly This journal is © The Royal Society of Chemistry 2018 View Article Online Open Access Article. Published on 05 April 2018. Downloaded on 17/05/2018 12:46:17. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. Edge Article underlined by Lunkenheimer and Köhn in their work describing the coupling of the ADC(2) theory to a continuum approach of solvation effects.26 As another alternative to TD-DFT, the Bethe–Salpeter equation (BSE) formalism31–35 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-transfer36–44 and cyanine-like45,46 excitations, while preserving the same O ðN 4 Þ 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 calculat (...truncated)