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WIREs Comput Mol Sci
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Going beyond the vertical approximation with time‐dependent density functional theory

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Since two decades, time‐dependent density functional theory (TD‐DFT) has been in the limelight due to its noteworthy efficiency. Indeed, in many cases, TD‐DFT provides accurate excited‐state properties for a relatively limited computational cost. Recently, there has been a rapidly growing interest in applications of TD‐DFT (partly) exploring the excited‐state potential energy surfaces. In this review, we summarize such TD‐DFT investigations going beyond the vertical approximation and devoted to spectroscopic applications, with a focus on both 0–0 energies and vibrationally resolved absorption and emission spectra. We show how these quantities can be computed, considering various models for the latter, and illustrate their advantages compared to vertical estimates in terms of comparisons with experimental data. WIREs Comput Mol Sci 2016, 6:460–486. doi: 10.1002/wcms.1260

Simplified energy diagram representing only two singlet states without intersections and describing the key theoretical parameters used here. For the same two potential energy surfaces (PESs), the adiabatic parameters and harmonic model PES are displayed on the left hand side, whereas, vertical values and harmonic model PES are shown on the right hand side.
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Electronic circular dichroism spectrum of the 1Lb transition α‐methyl‐phenylethane computed at 0 K with FCHT|AH model on the basis of TD‐CAM‐B3LYP/TZVP data, compared to the experimental spectrum at 183 K. Computed spectrum was convoluted with a Gaussian with FWHM = 320 cm−1 and redshifted by 4650 cm−1 to ease the comparison with the experimental shape. Reprinted with permission from Ref. . Copyright 2013 American Chemical Society.
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TPA spectrum of p‐nitro‐aniline computed at TD‐DFT level with different XCFs and with CC2 method adopting the cc‐pVTZ atomic basis set. The blue, purple, green and red curves respectively correspond to the CC2, BHHLYP, CAM‐B3LYP and B3LYP data. Together with the total spectra, the FC and HT pure contributions and the FC/HT interferential contribution have been reported. Reprinted with permission from . Copyright 2015 WILEY‐VCH Verlag GmbH.
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One photon (OPA) and two‐photon (TPA) absorption of the first (S1) and second (S2) excited‐states of a dioxaborine heterocyclic compound computed at PCM‐TD‐B3LYP/6‐31G(d) level in dichloromethane within the FCHT|AH model (also pure FC and HT contributions are shown) and convoluted with a Lorentzian with FWHM = 1600 cm−1. Dashed vertical lines show the vertical excitation energies. In addition, the experimental OPA spectrum is reported for comparison. Reprinted with permission from Ref. . Copyright 2011 WILEY‐VCH Verlag GmbH.
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Experimental (black lines) and theoretical (red lines and blue sticks) band shapes for two substituted anthraquinones. To allow direct comparisons, the theoretical 0–0 energies have been shifted to the experimental values. The experimental colors are compared to their theoretical counterparts (obtained from vertical TD‐DFT values or vibronic calculations) on the rhs of each graph. Reprinted with permission from Ref. . Copyright 2012 American Chemical Society.
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Comparison between experimental (full black lines) and B3PW91/LanL2DZ theoretical (full and dashed red lines) phosphorescence spectra for two inorganic complexes. The dashed theoretical spectra is shifted to match the experimental energies of the first band. Reprinted with permission from Ref. . Copyright 2015 Springer.
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Comparison between experimental and theoretical 0–0 energies for 185 fluoroborate fluorophores. All results have been obtained with a TD‐M06‐2X/6‐311+G(d,p) approach accounting for solvent effects through the polarizable continuum model. Top: Linear‐Response (LR) formulation for (PCM) solvent effects. Bottom: SS‐PCM formulation. These data include 92 BODIPY, 13 aza‐BODIPY, 31 NBO dyes, 25 OBO compounds, and 24 ladder‐type fluoroborates. We refer the readers to these original references for molecular structures, list of solvents used and experimental references.
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S0S1 absorption spectrum of trans‐azobenzene computed at 0 and 373 K at the TD‐PBE0/6‐31G(d) level adopting the FC|AH model and a convolution with a Gaussian with FWHM = 320 cm−1, compared with the experimental vapour phase spectrum at 373 K. Reprinted with permission from Ref. . Copyright 2013 American Chemical Society.
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Emission spectrum of free base porphine computed at FC|AH and FCHT|AH level of theory and convoluted with a FWHM = 80 cm−1. Reprinted with permission from Ref. . Copyright 2014 AIP Publishing LLC.
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S0S1 absorption spectrum of coumarin C153 in gas‐phase computed at the TD‐PBE0/6‐31G(d) level with different Adiabatic (A) or Vertical (V) models for the PES and using either the FC or the FCHT approximation for the transition dipole. The latter, in HT approximation, is expanded around the initial state equilibrium geometry (HTi) in combination with VH model and around the final‐state equilibrium geometry (HTf) with the AH model. Spectra convoluted with a Gaussian with FWHM = 800 cm− 1. Reprinted with permission from Ref. . Copyright 2012 PCCP Owner Societies.
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S0S1 absorption spectrum of HBDI computed with TD‐CAM‐B3LYP, CAS‐SCF,CAS‐PT2 and XMCQD‐PT2 (XMCQ) and the 6‐31G(d) basis set with the FC|AS model and convoluted with a Gaussian with FWHM = 320 cm−1. Reprinted with permission from Ref. . Copyright 2014 WILEY‐VCH Verlag GmbH.
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Comparison between experimental and theoretical band shapes (relative intensities) for the absorption spectrum of coumarin NKX‐2586. Spectra computed with the FC|AH model at room temperature with PBE0 and CAM‐B3LYP functionals and the 6‐31G(d) basis set are reported both excluding (label ‘hr’, high‐resolution, convoluted with a narrow Gaussian with FWHM = 160 cm−1) and including the solvent broadening estimated with SS‐PCM/TD‐DFT (label ‘PCM’). Figure based on data reported in Ref. .
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MSE (top) and MAE (bottom) obtained for the relative positions of the maxima in the absorption (left) and emission (right) bands of 20 compounds. All values have been obtained by setting the position of the TD‐DFT 0–0 band to the experimental value and determining the differences of positions for the other maxima. All data are in cm−1.
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Comparison between experimental and theoretical (FC|AH) band shapes for an absorption spectrum (top) and a fluorescence spectrum (bottom) of two organic compounds. The 0–0 band was set to 0 cm−1 and the maximal intensity set to 1, in both theoretical and experimental spectra to allow straightforward comparisons. Reproduced with permission from Ref. . Copyright 2013 American Chemical Society.
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