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Excited State Dipole Moments in Solution: Comparison Between State-Specific and Linear-Response TD-DFT Values Ciro Achille Guido, Benedetta Mennucci, Giovanni Scalmani, and Denis Jacquemin J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01230 • Publication Date (Web): 31 Jan 2018 Downloaded from http://pubs.acs.org on February 2, 2018

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Journal of Chemical Theory and Computation

Excited State Dipole Moments in Solution: Comparison Between State-Specific and Linear-Response TD-DFT Values Ciro Achille Guido,∗,†,‡ Benedetta Mennucci,¶ Giovanni Scalmani,§ and Denis Jacquemin∗,† †Laboratoire CEISAM - UMR CNRS 6230, Université de Nantes, 2 Rue de la Houssinière, BP 92208, F-44322 Nantes Cedex 3, France ‡Laboratoire MOLTECH - UMR CNRS 6200, Université de Angers, 2 Bd Lavoisier, F-49045 Angers Cedex, France ¶Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Via G. Moruzzi 13, I-56124 Pisa, Italy §Gaussian, Inc., 340 Quinnipiac Street, Building 40, Wallingford, Connecticut, 06492, USA E-mail: [email protected]; [email protected]

Abstract We compare different response schemes for coupling continuum solvation models to Time-Dependent Density Functional Theory (TD-DFT) for the determination of the solvent effects on the excited state dipole moments of solvated molecules. In particular, Linear-Response (LR) and State-Specific (SS) formalisms are compared. Using twenty low-lying electronic excitations, displaying both localized and charge-transfer character, this study highlights the importance of applying a SS model not only for the calculation of energies, as previously reported (J. Chem. Theory Comput., 2015, 11, 5782), but

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also for the prediction of excited state properties. Generally, when a range-separated exchange-correlation functional is used, both LR and SS schemes provide very similar dipole moments for local transitions, whereas differences of a few Debye units with respect to LR values are observed for CT transitions. The delicate interplay between the response scheme and the exchange-correlation functional is discussed as well, and we show that using an inadequate functional in a SS framework can yield to dramatic overestimation of the dipole moments.

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Introduction

The accurate simulation of geometries, energies and properties of electronic excited states (ES) is fundamental for the understanding of many photochemical and photophysical processes of technological or biological interest. The efficient exploration of the ES potential energy surfaces of molecular systems has received a substantial boost from the formulation of analytical first 1–3 and second 4,5 derivatives of Time-Dependent Density Functional Theory 6 (TD-DFT) in the random phase approximation (RPA) formalism. 7,8 Such an approach shows an advantageous ratio between computational cost and accuracy. Nevertheless, the RPA formulation of TD-DFT inherits the typical problems of ground-state (GS) DFT together with those originating from the linear-response (LR) and adiabatic approximations. These two approximations do not significantly affect the calculation of electronic transitions involving valence electrons (also called locally excited states, LE) which, as extensively shown in literature (see, for instance, Refs. 9–13), are reproduced with a good accuracy. In contrast, benchmarks showed that the LR-TD-DFT approach, combined with approximate exchange-correlation functionals (XCF), has difficulties in describing charge transfer (CT) excitations, multielectron excitations, and, more in general, absorption spectra of systems with delocalized or unpaired electrons. 13–16 To address these issues, at least the ones related to CT or Rydberg transitions, the introduction of long-range corrections, as formulated in the so-called range-separated hybrid (RSH) XCF greatly improves the accuracy. 17–20 2


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Despite these methodological progress, the usefulness of all these computational approaches remains quite limited if environmental effects are not accounted for. Indeed, almost all photochemical and photophysical processes of interest take place in presence of an environment, e.g., a solvent, a protein matrix, a thin solid film or a crystal, that can significantly tune the properties of the system. The large majority of the computational models currently available rely on a focused approach, i.e., they combine a Quantum Mechanical (QM) description of the molecular system of interest (the chromophore, possibly including small portions of the environment) to a classical description of the remainder. 21 There are different formulations of such focused approach: the most common ones are obtained by combining a QM description with Molecular Mechanics (QM/MM), possibly including polarization in the force fields, or continuum solvation models. If specific interactions between the solute and specific solvent molecules do not play a fundamental role, continuum models are certainly more effective than QM/MM to include environmental effects. In particular, a very effective formulation of continuum models is based on the use of an apparent surface charge (ASC) distribution to represent the polarization of the dielectric medium. 22,23 As a result the whole response of the solvent is limited to the surface of the molecular cavity embedding the QM solute. In continuum models, the study of electronic transitions introduces two new complications. First, one needs to describe a correct solvation regime: when a fast change in the electronic density of the solute occurs, e.g., during a vertical electronic transition, only the electronic component of the solvent polarization responds on the same timescale, while the rest (the inertial component) remains frozen in a configuration which is still in equilibrium with the initial solute charge density. In models based on a ASC, such non-equilibrium regime is modeled by employing two sets of surface charges: a dynamic one, which is equilibrated with the new solute charge density and an inertial one, which is frozen in equilibrium with the initial conditions. The second complication introduced by continuum models comes from the polarization scheme used. Within the ASC formulation, the specificities of the extension

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of QM solvation models to describe excitation processes have been rigorously analyzed from both formal and numerical points of view. 24,25 This analysis has shown that the application of a continuum model either as a Linear Response or a State-Specific (SS) correction to TD-DFT or CIS (Configuration Interaction Singles) electronic structure methods may lead to differences in the description of the ES, due to the intrinsic nonlinear character of the solvent response operators. The same considerations hold also for discrete polarizable models, based on a classical description of the solvent. 26,27 The SS methods, which require the explicit calculation of the ES wavefunction, properly take into account the variation of the solvent polarization associated to the change in the solute’s electronic density during an electronic excitation, whereas the LR approach 28,29 introduces an effect related to the transition density only. This latter effect can be viewed as a dispersion-like contribution 25 or as an excitonic coupling term. 30 Three different schemes have been proposed to achieve a SS description of the solvent response within the context of TD-DFT, namely (a) the corrected linear response (cLR), 31 (b) the self-consistent approach discussed by Improta, Barone, Scalmani and Frish (IBSF), 32 and (c) the Vertical Excitation Method (VEM). 33 All three schemes have been implemented for the Polarizable Continuum Model (PCM) either in the Integral Equation Formalism (IEF-PCM) 34 or the Conductor-like (C-PCM) 35 formulation, and involve corrections that depend on the TD-DFT ES density. However, whereas the cLR and VEM schemes make use of the excitation contribution to the one particle density matrix, that accounts for the orbital relaxation effects, the IBSF scheme introduces a dependence on the total ES density, therefore modifying also the GS density. More in details, cLR provides a perturbative correction only, and is therefore not self-consistent, which differs from both IBSF and VEM. The VEM, in particular, can be formulated in two ways: one that requires only the unrelaxed ES density (UD), and another where the relaxed density (RD) is used. Recently, the VEM approach in the UD formulation, has been made more useful by the derivation and implementation of the analytic first derivatives of the ES energy. 36,37 The pros and cons of these three different

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polarization schemes have been carefully studied in the case of transition energies, 38–42 and the main result of these studies was that a SS approach is essential to properly describe ES showing strong CT character. However, much less attention has been devoted until now to ES properties. The first two studies, focused on the impact of cLR on ES geometries 43 and vibrations, 44 have been performed using numerical differentiation of cLR energies of small chromophores. More recently, the analytical gradient of the TD-DFT/VEM-UD energy, 37 based on a Lagrangian formulation, has been used to calculate ES properties and structures of chromophores in solution according to an SS approach. The applications to few selected cases showed that, when the electron density reorganization is negligible, the geometrical changes predicted by the VEM-UD model are bracketed by their gas phase and LR counterparts, while they become much more relevant for electronic transitions involving a large electron density rearrangement. In the present contribution, we extend the analysis of the behavior of the different polarization schemes to ES properties. More specifically, as an indicator of the impact of solvent effects on the ES density, we have chosen the ES dipole moment, which is a property of great importance for applications, but hard to measure experimentally. Here we have considered the ES dipole moments of twenty molecules in solution, focusing on the lowest-lying excitations (14 LE and 6 CT). We have obtained analytical values with both LR and VEM-UD schemes, together with cLR numerical values, and assessed the impact of solvent polarity, of the solvation regime (equilibrium vs non-equilibrium) and of the XCF being used on the computed dipoles. The delicate interplay between the polarization method and the XCF is considered, providing an assessment going beyond the one previously reported for ES energies. 38

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Computational details

All the TD-DFT calculations have been performed using the aug-cc-pVDZ atomic basis set, and two global hybrid (GH) exchange-correlation functionals, namely PBE0 45,46 and PBE0-1/3, 47 one meta-GGA GH functional, M06-2X, 48 and two RSH functionals, i.e., LCωPBE 49,50 and ωB97X. 51 Solvent effects have been introduced by means of the IEF-PCM 34,52 approach, with a molecule-shaped cavity build from interlocking spheres centered on all atoms and sized using the UFF radii. We employed the high accuracy ultrafine integration grid (which comprises 99 radial shells and 590 angular points per shell) to ensure the numerical stability of our results. We have used a development version of the Gaussian program, 53 that includes the analytic implementations of the VEM-UD energy and gradient, and we made the modifications necessary to perform numerical derivatives of the cLR energy with respect to an applied electric field to compute the cLR dipole moments. All dipole moments have been determined for molecular systems at their gas-phase GS optimized geometry, as obtained at the M06-2X/6-31+G(d) level of theory. The Cartesian coordinates of all systems are available in the Supporting Information (SI) of Ref. 54.

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Results and discussion

The twenty medium-sized organic molecules included in this study are depicted in Scheme 1. They were chosen to be representative of compounds of interest for photoactive applications. Molecules of the first set (1–14) display LE transitions, whereas compounds in the second set (15–20) are characterized by CT excitations, whose description is known to be challenging for TD-DFT, when standard semi-local XCF or GH functionals with a low percentage of exact-exchange (EXX) are used. All excitations are bright and can be essentially described as HOMO→LUMO transitions. Indeed, all molecules studied in this contribution are experimentally known and their optical spectra have been measured in solution (see Ref. 54 for references). In the present work, we focus on the comparison of different PCM-TD-DFT 6


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response schemes, considering both in the equilibrium and non-equilibrium solvation regimes. As representative apolar and polar solvents, cyclohexane and acetonitrile have been chosen, respectively. Our results for the GS dipole moments are given in Table S1 of the SI, whereas all ES data discussed below are listed in Tables S2–S5 of the SI. We note that, as the GS dipole moments are the same for LR, cLR and VEM-UD approaches, differences between ES dipoles computed with these three methods (Tables S4 ans S5) are also differences between the GS and ES dipoles determined with these approaches.

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Scheme 1: Structures of the molecules considered for this study. Compounds 1–14 show LE excitations whereas compounds 15–20 are characterized by CT excitations. Recently, 38 we have discussed the risk of overpolarization effects when an inadequate XCF is used to describe transitions involving large density rearrangements: the nature of the XCF determines the strength of the polarization induced by the transition density as well as the magnitude of the change in density between the GS and the ES. The conclusion of that 7



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study was that in several cases the use of a RSH XCF together with a SS treatment is needed to achieve a physically-sound description of the transition. Therefore, in the following, we focus first on the results obtained with an RSH functional, LC-ωPBE, before considering other XCFs.

3.1

Excited state polarization response energies

Let us first discuss the formation of the excited state of a solvated molecule as a two-step process: i) the molecule in its GS and in equilibrium with the solvent is vertically excited to (0)

the i-th state in the presence of a frozen solvent polarization (let’s indicate ω0i the resulting change in energy); this approximation is also known as the Ground State Frozen Polarization (GSFP) reaction field; 33 ii) the response of the solvent is switched on and its polarization rearranges to equilibrate with the ES charge density of the solute. We will call the resulting (0)

correction to ω0i which accounts for this contribution polarization response energy RX , with X=LR or SS. Within this theoretical framework, the change in energy associated with the ground-to-excited state transition in the LR or the SS schemes can be written as:

(0)

(1)

(0)

(2)

LR ω0i = ω0i + RLR (X0i ), SS ω0i = ω0i + RSS (P0i∆ ),

and,

ˆ ˆ (0) i, RLR (X0i ) = hi(0) |V|0ih0| Q|i RSS (P0i∆ ) =

1 ˆ − h0|V|0i][hi| ˆ ˆ − h0|Q|0i], ˆ [hi|V|ii Q|ii 2

(3) (4)

where, for a generic excited state i, X0i represents the transition density whereas P0i∆ is the change in the density with respect to the ground state (in VEM-UD, this does not

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ˆ include the contribution from the molecular orbitals relaxation). In Eqs. (3) and (4), V ˆ is the operator that defines the is the molecular electrostatic potential operator whereas Q apparent surface charges. RX can be either formulated in the non-equilibrium or equilibrium ˆ provides either to the charges due to a solvation regime, and accordingly the operator Q

5.0

5.0

4.0

4.0 VEM-‐UD (eV)

cLR (eV)

dynamic (non-equilibrium) or static (equilibrium) response.

3.0

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0.0

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0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

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ΓNTO (Å)

ΓNTO (Å)

Figure 1: Correlation plots of the S1 excited state energies (blue: LE; red: CT). Top: S1 excited state energies (eV), cLR (left) and VEM-UD (right) values as a function of the LR ones. Bottom: cLR (left) and VEM-UD (right) S1 excited state polarization energies (eV), as a function of the effective electron displacement of the considered ES (ΓNTO in Å). All data were obtained in acetonitrile in the equilibrium solvation regime. In Figure 1 we report the correlation between the excitation energies calculated with one of the two SS schemes (cLR or VEM-UD) and those calculated with the LR scheme. All the values are calculated according to an equilibrium regime and using acetonitrile as solvent. The corresponding results for the apolar solvent and the non-equilibrium solvation regime are 9



reported in the SI. On average, for LE transitions the polarization response energy is -0.23 eV using LR, -0.03 eV with cLR, and -0.06 eV with VEM-UD, whereas, in the case of CT, the LR, cLR and VEM-UD mean values are -0.17 eV, -0.18 eV, and -0.42 eV, respectively. In other words, the differences between LE and CT states appear more clearly when applying the SS approaches than the LR model. Even if the impact of the specific response scheme is large, a linear correlation between the LR and the SS excitation values can still be observed. Indeed, the LR LE transition energies show a positive correlation for both cLR and VEMUD; this correlation slightly decreases in the case of CT excitations, see the top panels of Figure 1. As cLR is a perturbative correction to LR, that can be considered as the first iteration of a VEM-RD calculation, it is not unexpected that the correlation for CT states deteriorates when going from cLR to VEM-UD. We additionally note that the SS polarization response energies show a parabolic dependence with respect to the effective electronic displacement induced by the excitation, as estimated by the ΓNTO 55 metric, based on the use of the natural transition orbitals 56 (NTOs). ΓNTO is defined as the sum of the displacement and the spread around the centroid of charges described by the NTO. Indeed, in the bottom panels of Figure 1, the parabolic interpolation passing through the origin nicely fits the observed trend: the magnitude of the SS correction increases almost quadratically with the spatial reorganization of the density, at least when using the LC-ωPBE RSH functional.

3.2

Excited state dipoles

Let us now discuss the ES dipoles obtained using the different solvent response schemes. As stated above, in the VEM-UD variant of the SS scheme, the ES reaction field responds to the unrelaxed component of the ground-to-excited state density difference P ∆ , while it does not include the molecular orbital relaxation contribution. In order to estimate this contribution, we calculated the difference between the unrelaxed and relaxed ES dipoles obtained using a GSFP reaction field, i.e., the first step of the VEM iterative procedure, corresponding to 10


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(0)

the ω0i energy. On average the differences are ca. 0.5 D, and therefore we can expect that VEM-UD yields values very close to their VEM-RD counterpart, at least for the RSH XCF used here. The only exception is compound 20 for which the computed difference is as large as 2.4 D, and one can therefore expect that for this dye, VEM-UD overestimates the solvent response. The ES LR dipole moments and differences between SS and LR values are reported in the SI (Tables S4 and S5). Concerning the LE transitions, the mean absolute deviation (MAD) between SS and LR dipole moments is small, which is a pattern that is observed for both equilibrium and non-equilibrium solvation regimes and for both the cLR and VEM-UD methods. Indeed, the deviations are typically smaller than 0.5 D, with the exception of the calculations performed in acetonitrile with the equilibrium regime. In that case, the largest differences are observed for compounds 10 and 11 with changes compared to LR of ca. 1.5 D with cLR and 2.7 D with VEM-UD. As it can be expected, larger effects are observed for CT excited states. If one considers a non-equilibrium regime, the MAD is ca. 1.7 D, but larger differences around 5 D with VEM and 2–3 D with cLR, are observed for the zwitterion 16 as well as for compound 20. The effect of the SS correction becomes much larger switching to an equilibrium regime in a polar solvent: the MAD with respect to LR reaches 3.2 and 5.4 D, for cLR and VEM-UD respectively. Again, there are significant differences among the compounds under consideration, with compound 20 showing the largest effect (namely 7.6 D and 16.0 D with cLR and VEMUD, respectively). As for the polarization response energies, a linear correlation between the LR and the SS dipole values is observed for ES dipoles, especially for LE transitions; the correlation being slightly smaller for CT excited states, as shown in Figure 2. As a final comment to this Section, we would like to point out that the excitation energies are significantly dependent on the cavity size. 57 This effect is obviously enhanced for CT state and SS approaches, because the response of the reaction field is larger compared to LE transitions. In the SI (Figure S5), the ES energies and dipole moments obtained by

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increasing the scaling factor of the UFF spheres used to build the cavity (α parameter, controlled by the alpha= PCM keyword in Gaussian) from 1.1 (default value) to 2.0, for the CT state of compound 15 are reported. As expected, both LR and VEM-UD simulations tend to the gas-phase values when the cavity is enlarged, and hence the differences between the LR and VEM-UD dipoles decreases with increasing α. 45

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Figure 2: Correlation plots of SS S1 ES dipole moments (Debye) with respect to LR (blue: LE; red: CT). Left: cLR, right: VEM-UD. All values refer to the equilibrium solvation regime in acetonitrile.

3.3

Interplay between the XCF and the solvent polarization scheme

In this Section, we discuss the effects of changing the XCF and the interplay with the three response schemes considered in this work. Some key results are presented in Figure 3, where the difference with respect to the LC-ωPBE dipoles are reported. We discuss only the equilibrium regime in acetonitrile, as this combination yields the largest solvation effects. Generally, and irrespectively of the response scheme adopted, we note that ωB97X, which is a RSH functional with correct asymptotic behavior, provides essentially the same results as LC-ωPBE: the largest discrepancy does not exceed 1.7 D. Indeed, even if the exchange and correlation contribution of these two XCF are different, they both include 100% of EXX at long-range and this decisively affects the description of the ES. It is well-known that the description of the ES state is greatly dependent on the amount of EXX: 19,58 GGA 12


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and GH with low content of EXX tend to overstabilize CT states and exaggerate the density reorganization as the electron and the hole do not correctly interact when they do not overlap. It is therefore expected that the magnitudes of the ES dipole moments of CT states increase when the amount of EXX included in the XCF decreases. While this statement holds for all response schemes, it is evident from Figure 3 that the differences are enhanced going from LR to cLR and next to VEM-UD: the solvent response in the SS schemes is a function of the ground-to-excited state electron density difference (P∆ ), and this term is larger with GH than with RSH XCF. We stress that for 20, the VEM-UD, which is a self-consistent SS scheme, significantly increases the CT character of the ES even with RSH functionals. Consequently, the final ES dipole moment is very large, and the differences between the functionals becomes negligible. In that sense, the CT character is always “maximized” for that specific molecule when using the VEM-UD approach. We note that for 20, the system with the largest CT character, the amount of reorganization of density induced by the excitation is almost the same for all the functional used (see Figures S1 and S2 in the SI). Therefore this outcome is typical of this compound and is probably not related to an inaccurate VEM-UD description. Let us now focus on the LE transitions (compounds 1-14). In the LR formalism, very small changes in going from RSH to PBE0 are observed: the largest deviation are ca. 2 D for compounds 10–12. The situation is significantly different in the case of the SS models: the ES dipole moments of some compounds greatly increase when decreasing the EXX percentage in the XCF, an effect particularly remarkable for both the cLR and VEM-UD results of compounds 10–12, as well as the VEM-UD dipoles of compounds 2 and 6. This is a consequence of the stabilization of the CT electronic distributions due to combined effects of the use of a GH XCF and a SS solvent response. To better illustrate this effect, we have considered the properties of molecule 12 as given by the M06-2X and LC-ωPBE functionals. The first ES is described as local by both functionals when using the LR formalism. However, according to the VEM-UD formalism, two CT configurations start to contribute to the M06-2X description of this ES state and the final ES is a mix of a LE (35%) and CT (28%

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Figure 3: ES dipole moment difference (∆µ, Debye) with respect to LC-ωPBE values obtained for the 20 compounds with various XCF. From top left to bottom right: vacuum, LR, cLR and VEM-UD. and 13%) molecular orbital contributions (see Figure 4). This phenomenon does not occur when using a RSH, as shown by the final VEM-UD representation of the transition given by the NTOs in Figure 5. Similar effects were found for several other compounds (2, 6, 10, and 11). These results are in line with our previous findings related to transition energies: 38 the use of hybrid functional including a fixed amount of EXX in conjunction with a SS solvation approach can cause strong overstabilization of states with large electron density rearrangements, due to the synergy of the choice of XCF and the solvent response equilibrated to the ES density. Essentially, decreasing the EXX included in the XCF induces an increase of the 14


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H

H

H -1

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L

VEM-UD

92%

35%

0%

28%

0%

13%

L+1

L

Figure 4: MOs contributions to the S1 ES of system 12, obtained using the M06-2X functional. Equilibrium solvation regime in acetonitrile. M06-2X

LC-ωPBE

Figure 5: NTOs contributions to the VEM-UD S1 ES of system 12, obtained using the M06-2X and LC-ωPBE functionals. Equilibrium solvation regime in acetonitrile. polarizability of the electron density. The dipole moment of an excited state i (µi ) can be defined as the derivative with respect to an external electric field (F~ ) of the total ES energy (Ei = E0 + ω0i ): µ ~i =

∂Ei ∂E0 ∂ω0i = + =µ ~ 0 + ∆~µ0i . ∂ F~ ∂ F~ ∂ F~

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(5)


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Making use of Eq. (2), we can write the following expression for a SS approach: (0)

µ ~ SS ~ 0 + ∆~µ0i (ω0i ) + i = µ

∂RSS [P0i∆ ] . ∂ F~

(6)

By using a very simplified model, namely a dipole in a spherical cavity, 25,31,38 one can extract further physical insight by writing:: 1 RSS [P0i∆ ] ≈ − f (ε, ε∞ , S)∆µ20i , 2

(7)

where f (ε, ε∞ , S) is the reaction field factor for a dipole in a spherical cavity of radius S. Therefore, as a first approximation, we can evaluate the SS contribution to the ES dipole as:

δRSS =

∂RSS [P0i∆ ] ≈ −∆~µ0i ∆α0i f (ε, ε∞ , S) ∂ F~

(8)

where ∆α0i is the ground to excited state variation of the static polarizability. Therefore, following Eq. (6), the values in Figure 3 are the results from a sum of different contributions: (0)

∆µSS µ0 + ∆∆~µ0i (ω0i ) + ∆δRSS LC−ωPBE−XCF = ∆~

(9)

where the different terms are obtained from differences of the contributions obtained through Eq. (6) with LC-ωPBE and another XCF. We calculated ∆µ0i and ∆α0i in a GSFP framework, in order to analyze the behavior of the different terms. We choose M06-2X as a representative example. In the left panel of Figure 6 all the terms in Eq. (9) are compared: in orange the change in GS dipole, in green the ground-to-excited dipole moment change, in red, the specific SS term. In addition, we have also plotted in blue, the approximate version (0)

(0)

of the last term defined in Eq. (8), that is, ∝ ∆[∆µ0i (ω0i )∆α0i (ω0i )]. Even if this is a very rough model, the qualitative correlation between the approximate and analytical terms is obvious. We also highlight that, even in the case of a GSFP approach, there is a significant

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difference of ca. 1.3 D for the same LE compounds that were found very sensitive to the use of the VEM-UD approach. This is in fact due to the effect of a given XCF has on the static polarizability of the electron density, as clearly shown in the right panel of Figure 6, for the (0)

∆∆α0i (ω0i ) term. As a final comment, we would like to point out that the molecular orbital relaxation contribution (the so-called Z-vector) to the ground-to-excited state density difference P ∆ becomes larger when decreasing the EXX percentage included in the XCF (see the SI, where a comparison of the two contributions to the ES dipoles in gas phase is reported). This orbital relaxation contribution usually leads to a reduction of the ES electron density rearrangement following the transition. As in VEM-UD the ES reaction field responds just to the unrelaxed component of P ∆ , the origin of the overpolarization effects observed for GH functionals is due to an overestimation of the ES electron density rearrangement. Therefore, it is essential to check this difference, at least in gas-phase, in order to conscientiously apply the VEM-UD approach for ES dipole calculations. In a sense, the solvent response scheme can act as a probe of the well-known CT overstabilization problem of GH in TD-DFT: the unsuitability of the XCF can be established more easily when a SS solvation approach is used. 4.0 3.5

Δµ0 Δμ(GS)

ΔΔµ0i(ω0) ΔΔμ(T+Z)[ω0]

500

ΔδR(T)[VEM] ΔδRVEM

Δ[Δμ(T) x Δ0)Δα α] 0i(ω0)] Δ[Δµ0i(ω

450 400

2.5

350

ΔΔα (a. u.)

3.0

2.0 atomic units

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.5 1.0

-‐1.5

200

100

0.0

-‐1.0

250

150

0.5

-‐0.5

300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

50 0

Compound

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Compound

Figure 6: Left: Comparison of the different terms involved in the difference between M06-2X and LC-ωPBE ES dipole moments, as given in Eq.(9). Right: Difference between M06-2X and LC-ωPBE ES isotropic polarizability difference (∆∆α). All values are in atomic units and have been obtained in acetonitrile with the equilibrium regime.

17



3.4

Further discussion

The main goal of the study presented is the critical comparison of LR and SS schemes to describe ES dipole moments. Nevertheless, it is also interesting to compare the performances of the different approaches to experimental results. We note that the experimental values of ES dipole moments are generally obtained through indirect procedures, that is, often by studying solvatochromic effects and extrapolating the results using simplified models, such as spherical cavity. Therefore, a straightforward quantitative comparisons between theoretical and experimental values should be made with caution. Nevertheless, to offer some insight from computation, we report in Table 1 the ωB97X-D/aug-cc-pVDZ dipole moments of the CT transitions of two dyes of practical use, namely DMANS and C153 (see Scheme 2) for which experimental values have been reported in toluene. 59 In these cases, the VEM-UD approach clearly provides an improved description of the ground-to-excited state variation of dipole moments compared to the usual LR approach, especially for DMANS. Table 1: ωB97X-D/aug-cc-pVDZ GS (µGS ) and ground-to-excited state variation (∆µES ) of dipole moments of systems DMANS and C153 in toluene. All values in Debye. Experimental values from Ref. 59. µGS Compound IEF-PCM Exp. DMANS 10.8 7 C153 8.9 6.5

LR 12.9 6.5

∆µES VEM-UD Exp. 23.9 24 7.6 9

Scheme 2: Representation of DMANS and C153.

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4

Conclusions

We have used the excited state dipole moment as a metric to investigate the impact of different solvent response schemes on the electronic density of the lowest-lying LE and CT states of twenty compounds. Both LR and two different formulations of SS response schemes (VEM-UD and cLR) have been compared. For both LR and VEM-UD schemes, the dipole moments have been computed analytically while for cLR a finite-difference approach has been used. The impact of solvent polarity, solvation regime (equilibrium vs non-equilibrium) and XCF has been assessed. Our results indicate that, if a RSH with 100% of EXX at longrange is used (e.g., LC-ωPBE or ωB97X), the deviations between LR and SS dipoles tend to be small for LE transitions; a conclusion valid in both solvation regimes, both solvents and for both cLR and VEM-UD methods. A clear correlation between SS and LR dipoles is also found. Larger effects were highlighted for CT ES, particularly in a polar solvent with an equilibrium solvation regime: the mean absolute difference between LR and cLR (VEM-UD) ES dipoles being about 3.25 (5.44) D, with significant differences between the various compounds. A SS approach is therefore required to accurately describe the ES dipoles of CT states. In addition, a comparison of the results obtained with three GH XCF confirms what we previously established for excitation energies: the use of a functional including a low amount of EXX in conjunction with a SS solvation scheme can yield very strong overstabilization of states characterized by large electron density rearrangements, due to the synergy between the behavior of the XCF and the fact that the solvent response is equilibrated with the ES density. Differences in ES densities given by two XCF, which are almost undetectable with a LR solvation approach, become evident when using VEMUD. This work therefore confirms the necessity of using both a RSH functional and a SS solvation response scheme to quantify the solvent effect of transitions involving large density reorganization, a conclusion also supported by comparisons with experimental data, whereas, for LE transitions, a linear correlation exists between LR and SS ES dipoles. This latter result indicates that the LR values can be safely used for bright transitions involving a small 19



electron density variation during the excitation.

Acknowledgement C.A.G. and D. J. acknowledge the support of the RFI LUMOMAT and the Région des Pays de la Loire for financial support in the framework of the FCPolResp and EE-Fate projects. The collaboration between the French and Italian teams is supported by the CNRS through a PICS action (SodasPret). This research used resources of i) the GENCI-CINES/IDRIS; ii) CCIPL (Centre de Calcul Intensif des Pays de Loire); iii) a local Troy cluster; and iv) HPC resources from ArronaxPlus (grant ANR-11-EQPX-0004 funded by the French National Agency for Research).

Supporting Information Available GS dipole moments obtained with PBE0, PBE0-1/3, M06-2X and ωB97X functionals. ES energies and dipole moments obtained with the same four functionals using the three solvent polarization response schemes considered, both in the equilibrium and nonequilibrium solvation regimes. Correlation plots. Plot of the ES unrelaxed-to-relaxed dipole moment difference. Evolution of the polarizability difference as a function of EXX for 6 and 12. Evolution of the transition energy and ES dipole as a function of the cavity size for 15.

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Graphical TOC Entry Linear Response

VEM-UD

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Excited State Dipole Moments in Solution ... - ACS Publications

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