Exploring excited states using Time Dependent Density Functional Theory and density-based indexes

Coordination Chemistry Reviews 304–305 (2015) 166–178

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Coordination Chemistry Reviews journal homepage: www.elsevier.com/locate/ccr

Review

Exploring excited states using Time Dependent Density Functional Theory and density-based indexes Carlo Adamo a,b , Tangui Le Bahers c , Marika Savarese a,d , Liam Wilbraham a , Gregorio García e , Ryoichi Fukuda f , Masahiro Ehara f , Nadia Rega g , Ilaria Ciofini a,∗ a

PSL Research University, Institut de Recherche de Chimie Paris IRCP, CNRS–Chimie ParisTech, 11 rue Pierre et Marie Curie, F-75005 Paris, France Institut Universitaire de France, 103 Boulevard Saint Michel, F-75005 Paris, France c Université de Lyon, Université Claude Bernard Lyon1, Centre National de Recherche Scientifique, ENS Lyon, 46 allée d’Italie, 69007 Lyon Cedex 07, France d Drug Discovery and Development, Istituto Italiano di Tecnologia, Via Morego 30, I-16163 Genova, Italy e Department of Chemistry, University of Burgos, Plaza Misael Ba˜ nuelos s/n, 09001 Burgos, Spain f Institute for Molecular Science and Research Center for Computational Science, Nishigo-naka 38, Myodaiji, Okazaki 444-8585, Japan g Dipartimento di Chimica ‘Paolo Corradini’, Università di Napoli Federico II, Complesso Universitario di M.S.Angelo, via Cintia, 80126 Napoli, Italy b

Contents 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The density-based index for the description of ES nature and evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density-based indexes as descriptors for TD-DFT performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of DCT to describe and quantify the charge transfer at the excited state in molecular compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density-based indexes for the description of excited state evolution and reactivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a r t i c l e

i n f o

Article history: Received 8 December 2014 Received in revised form 16 March 2015 Accepted 16 March 2015 Available online 17 April 2015 Keywords: Excited states exploration Density-based indexes TD-DFT ESPT Charge transfer

a b s t r a c t The recent advances in the development and application of density-based indexes for the description of the nature and the quantification of the extent of charge transfer associated with a given electronic transition are here reviewed. Starting from the basic definition of the indexes, a brief overview of their potential as indicators of potentially problematic cases in the description of charge transfer excitations using Time Dependent Density Functional Theory (TD-DFT) will be first given together with their possible application for comparing TD-DFT results to post Hartree–Fock (post-HF) calculations. After this methodological part, several examples of the application of density-based indexes to describe, from a quantitative and qualitative point of view, the charge transfer character (for instance in push–pull systems) or to map excited state reaction pathways (for instance in the case of Excited State Proton Transfer reactions) will be given to exemplify the insights that these indexes may bring to the description and design of new compounds of potential technological relevance. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Recent developments in quantum mechanical methods have enabled the description of Excited State Potential Energy Surfaces (ES-PES) and have opened the route to many investigations aimed

∗ Corresponding author at: IRCP UMR 8247, 11 rue P et M Curie, F-75231 Paris, Cedex 05, France. Tel.: +33 144276728.. E-mail address: ilaria.ciofi[email protected] (I. Ciofini). http://dx.doi.org/10.1016/j.ccr.2015.03.027 0010-8545/© 2015 Elsevier B.V. All rights reserved.

166 167 170 171 173 177 177 177

at the description, and prediction, of several different excited state phenomena of both fundamental and more applicative relevance. Many of these studies make use of Density Functional Theory (DFT) rooted approaches [1] and more specifically of Time Dependent-DFT (TD-DFT) protocols [2], mainly due to their extremely favorable accuracy to cost ratio [3]. Nonetheless, these approaches suffer from several identified flaws in their description of excited state energetic and nature, mainly ascribable to the underlying exchange correlation functional used, that call for the setup of internal indicators of their accuracy. On the other hand,

C. Adamo et al. / Coordination Chemistry Reviews 304–305 (2015) 166–178

the alternative presented by post-Hartree–Fock methods (although having indeed increased significantly in efficiency in the last years so to enable the description of relatively complex systems) is still far more computationally expensive than TD-DFT and still somehow suffers from the lack of simple, yet chemically sound, descriptors of the computed excited state nature and character.[4] With the objective to develop a descriptor that is able to link between TD-DFT results and chemical intuition, several different indexes were recently developed primarily to spot potentially problematic descriptions provided at TD-DFT level. With this primary diagnostic aim, several descriptors, mainly computed from orbitals, were derived. In this context it is worth to mention the seminal work of Gritsenko and Baerends [5] followed by the development of the Tozer index [6,7] or, more recently, those derived by Guido and Adamo [8,9] or Assfeld and collaborators [10,11]. Indeed, even if these last two indexes were also used to define the spatial extent associated with a given electronic transition, the use of all these indexes to measure charge transfer is still very limited [8–11]. In this general context, we developed in 2011 an index, the so called DCT , with the primary aim of quantifying the spatial extent associated with a given electronic transition in order to be able, for instance, to compare the strength in charge separation obtained in different families of donor–acceptor push–pull dyes ([D–A]) [12]. This index, contrary to the previous ones, is fully computed from the density associated with the ground and the excited state, and for this reason is directly extendable to any quantum chemical method, DFT or post-HF based, thus enabling a direct comparison of the results obtained with different approaches [13,14]. The strengths and weaknesses of the index are indeed both related to its very simple conception. The DCT quantifies the CT distance associated with a given electronic transition simply as the distance between the barycenters of the density depleted and density augmented zones associated with the electronic transition. Therefore, contrary to the Charge Displacement Analysis for excited states proposed by Tarantelli et al. [15], it condenses the information on charge reorganization upon electronic excitation to a CT distance. Interestingly, the DCT index is very similar to the distance between the electron and hole probability distributions lately defined by Faber et al. using GW calculations [16]. Thus, the strength of DCT is its ability to directly convey the simple chemical picture that, upon excitation, an electron is transferred in a molecule from one group (or atom) to another as is, for instance, represented by the [D+ –A− ]* excited state formulation in the case of push–pull dyes. It thus condenses and quantifies the information that one may derive for the analysis of the orbitals involved in the most relevant one electron excitations and describes a given excited state directly obtained at quantum chemical level. For this reason, this index has been largely applied for the description of CT excitation strength and extent in many molecular systems ranging from simple molecules to Push–Pull systems and dyes (including transition metal complexes) used for various applications such as, for instance, Dye Sensitized Solar Cells, NLO materials, fluorescent probes and sensors [17–60]. Nonetheless, the main drawback of the index also relies on its simplicity when condensing the information. If, from a theoretical point of view, the index formulation has the clear advantage of making no arbitrary choice in the localization scheme adopted, indeed the DCT is, by construction, vanishing for symmetric systems since the barycenters of density depleted and incremented zones coincide. The index is thus unsuitable to describe all centrosymmetric molecules such as, for instance, antenna systems of great photophysical relevance. This point has been also clearly underlined by Sun et al. [61] in their study of the electronic properties (band gap) of conjugated oligomers.

167

Nonetheless, beside this important weakness, the original DCT index has indeed been extensively applied not only to characterize molecular systems but also, more recently to follow excited state reactivity [54]. To this aim, a modified version of the index has also been proposed to analyze the evolution of the Excited State Potential Energy Surface (ES-PES) in the case of Excited State Proton Transfer reactions (ESPT) [54]. This work demonstrates its potential interest in the definition of suitable reaction coordinates for reaction occurring at the excited state. In order to give a flavor of the possible application of the densitybased index in the description of excited state nature and evolution essentially in combination with TD-DFT calculations, the review is organized as follows: after a description of the density indexes developed (Section 2), different examples of their application are summarized in the following sections. First (Section 3), the more methodologically oriented applications of DCT as an indicator of possible pathologic cases for TD-DFT or to more easily compare TD-DFT results with post-HF methods is given. Next (Section 4) examples of relevance of its application to the description of excited state nature and character are provided and finally, in section 5, examples of application of density indexes in the description of the excited state behavior and reactivity are briefly reported. Finally, some general conclusions about current limitations and strength of the indexes are given (Section 6). 2. The density-based index for the description of ES nature and evolution Here we will briefly review the definition of the density-based indexes as derived in Ref. [12,54]. Let us define GS (r) and EX (r) as the electronic densities computed for the ground and the excited state, respectively. The density change associated with an electronic transition from the ground to the excited state is thus given by: (r) = EX (r) − EX (r) − GS (r)

(1)

Two functions, + (r) and − (r), can be defined collecting, respectively, the points in space where an augmentation or depletion of the electronic density upon absorption is produced. That is:



+ (r) =

 − (r) =



(r) if (r) > 0 0 if (r) < 0 (r) if (r) < 0 0 if (r) > 0

(2)

(3)



and − (r)dr = − + (r)dr. The barycenters (referred in the following as R+ and R− ) of the spatial regions associated with + (r) and − (r), can thus be defined and computed, for instance by discretizing them on a 3D grid around the molecule, as:

 r+ (r)dr R+ =  = (x+ , y+ , z+ )

(4)

 r− (r)dr R− =  = (x− , y− , z− )

(5)

+ (r)dr

− (r)dr

The original DCT index was thus defined as the spatial distance between these two barycenters of density distributions as:





DCT = R+ − R− 

(6)

Integrating over all space + (r) or − (r) additionally provides the net transferred charge (qCT ) associated with the transition. Additionally, from the DCT and qCT terms defined above, the norm of the

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variation in dipole moment between the ground and the excited state (ES−GS ) can also straightforwardly computed:





||ES−GS || = DCT

+ (r)dr = −DCT

− (r)dr = DCT qCT

(7)

This value is clearly identical to the difference between the norm of the dipole moments computed for the ground and the excited states. In order to provide a measure of the spatial dispersion of the density increase and depletion zone around the barycenters, two centroids of charge associated with these positive and negative density regions were defined. To this end, from the root mean square deviations along the three axes ( aj j = x,y,z, a = + or −, Eq. (8)), the two centroids (C+ et C− ) were defined as described in Eqs. (9) and (10), respectively.



aj =

(r)(j − ja )dr

 C+ (r) = A+ e

qCT −

 C− (r) = A− e

−

(x − x+ )2 2 2+x

(x − x− )2 2 2−x

−

−

(y − y+ )2 2 2+y

(y − y− )2 2 2−y

−

−

(z − z+ )2 2 2+z

(z − z− )2 2 2−z

A+ =



e − (x−x2+ ) − 2+x

A− =

+ (r)dr

2



 2

−

(z−z+ )2 2 2+z

−

(z−z− )2 2 2−z

− (r)dr

e − (x−x2− ) − 2−x

(y−y+ )2 2 2+y

(y−y− )2 2 2−y



(14)

In analogy with Eqs. (2) and (3), +,react (r) and −,react (r) are defined as



+,react (r) =

 −,react (r) =



if (r) > 0

0

if (r) < 0



if (r) < 0

0

if (r) > 0

(15)

(16)

From Eq. (15) and (16) the barycenters (Rreact + and Rreact − ) of these difference density distribution can be computed and the DCT,react defined as:

(9)

+ − DCT,react = |Rreact − Rreact |

  (10)

(11) dr



(r)CT,react = EX,i (r) − GS,min S1 (r)

(8)

The normalization factors (A+ and A− ) were chosen so as to impose the condition that the integrated charge on the centroid equals the corresponding density change integrated over the whole space, that is



In particular, in the case of DCT,react , the ground state density (hereafter GS,min S1 (r)) is the one computed vertically from the minimal energy structure of the excited state PES considered while excited state density (EX,i (r)), is defined as the density associated with a given point (i) on the excited PES. As a consequence, (r)CT,react is defined as:

(12) dr

Beside their numerical value of DCT , both the density difference () the position of the barycenters (R+ and R− ) and the associated ellipsoids (C+ and C− ) can be easily visualized offering a suitable kit of tools for the interpretation of charge reorganization upon an electronic excitation as depicted in Fig. 1 in the case of a very simple push–pull system. In order to more easily define the through space character of a transition, an index (named H) was defined as half of the sum of the centroid axes along the maximum dispersion direction. For instance, if the maximum dispersion direction is x, H is defined by the relation: +x + −x H= (13) 2 If H ≥ DCT , this implies a substantial overlap between the centroids along the chosen direction. On the contrary, if H ≤ DCT the transition can be defined as “through space” since no overlap between density depletion and increase zones is computed. A general quantification of this through space character can thus be provided by the t index defined as DCT −H. Following the same philosophy originally used to develop the DCT , an incremental index (DCT,react ) was more recently proposed [54] offering a tool to track the evolution of a reaction occurring at the excited state. This index considers both the structural and the electronic reorganization associated with a charge transfer event. Note that, as in the previous cases the index is defined as the distance between the barycenters of two different density functions (+,react (r) and −,react (r)) whereas these latter indexes are no longer related to a vertical electronic excitation.

(17)

Physically, the DCT,react value quantifies how much the density changes (i.e. is transferred) from a given stable excited state nuclear conformation along the excited state PES. It follows that it implicitly takes into account both the electronic reorganization upon transition (actually already contained in the original DCT index) and the structural reorganization occurring from the excited state minimum to a given point at the excited state, as schematically depicted in Fig. 2. Both DCT and DCT,react are purely density-based indexes. However, it should be mentioned that it is possible to use Partial Atomic Charge (PAC) models such as Mulliken, Hirshfeld, Bader, Natural, Merz-Kollman and ChelpG, to compute the CT index [62]. Indeed, it is possible to define for each atom (i) whose Cartesian coordinates are ri = (xi ,yi ,zi ), the difference of PAC computed at excited and ground state: ıqi = qES − qGS i i as: q+ i

And to define two collections of atoms, respectively, q+ and q− i i

 =

 q+ i

(18)

=

ıqi

if

ıqi > 0

0

if

ıqi < 0

ıqi

if

ıqi < 0

0

if

ıqi > 0

(19)

(20)

The charge transferred upon excitation can thus be defined as: CT

q

=



q+ ≡− i



q− i

(21)

And, analogous to the case of density-based indexes it is possible to define the barycenters of these distributions of charges as:

r + = (x+ , y+ , z + ) =

r q+ i i i qCT

(22)

r q− i i i CT q

(23)

r − = (x− , y− , z − ) =

The distance between these two barycenters provides the charge transfer length: dCT = |r + − r − |

(24)

A simplified version of the PAC-based approach was more recently proposed by Adachi and collaborators [63], substituting the global charge computed at the ground and the excited state with

C. Adamo et al. / Coordination Chemistry Reviews 304–305 (2015) 166–178

169

Fig. 1. Computed (from Ref. [12]) difference in total density at the ground and excited state ((r) = EX (r)−GS (r), isocontour value 0.001 a.u.), graphical representation of DCT and R+ /R- and of the associated centroids of charge (C+(r)/C−(r), isocontour value 0.001 a.u.).

the percentage contribution of the different molecular fragments to the Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied MO (LUMO). Nonetheless, it has been shown that, although PACs based on a fit of the electrostatic potential seem to provide a CT distance closest to that obtained with a full density-based approach, no atomic charge model is fully equivalent in the prediction of the distance and charge associated with CT events, at least for the family of molecules tested [62,64], therefore their use is somewhat discouraged. In this context, it is also important to recall the works based on the Quantum Theory of Atoms In Molecules (QTAIM) to evaluate the CT extent highlighting the physical origin of these discrepancies between the PAC and full density models and to propose an alternative way to compute CT magnitude and extent based on QTAIM [64,65]. Nonetheless, it is worth stressing that all the examples provided in this review are based on the use of density-only based indexes.

The densities needed (for both the ground and the excited state) were computed on a grid of points around the molecule. In the case of the calculation of the DCT,react index a consistent grid definition and molecular orientation was chosen, namely shifting the center of mass and rotating the molecular axis in order to maximize the overlap between ground and excited state structures. As internal measure, the overall consistency of the size of the box used can be checked computing the net fraction of escaped electrons (ıesc ), that is:

 ıesc = N −

(r)dr

(25)

box

N being the total number of electrons in the molecule and (r) the total GS density. In the following all computed excited state densities are computed as described in Ref. [66] and references therein.

Fig. 2. Schematic representation of the ground and excited state density used to compute the DCT and DCT,react indexes.

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3. Density-based indexes as descriptors for TD-DFT performance

1000 900 800

λ (nm)

700

NH2 n

PBE PBE0 LCPBE CIS SACCI Exp

600 500 400 300 1

2

3

4

n 12 11 10 9

PBE PBE0 LCPBE CIS SACCI

8

DCT

7 6 5 4 3 2 1 1

2

3

4

n

t

At the moment of its proposal [12] one of the aims, though not the primary, of the DCT index was to probe the reliability of TDDFT based approaches in the prediction of excited state energy and character associated with charge transfer excitations. To this end a very simple family of rod-like Push–Pull dyes (the NH2 (phen)n NO2 , Fig. 3, hereafter referred to as Pn systems, n = 1–4) was first considered as test case. The transition energies and corresponding DCT indexes were computed using different exchange correlation functionals going from local (PBE) and global hybrid (PBE0) to range separated hybrids with correct asymptotic behavior (LC-PBE) and the results were compared with reference post-HF data computed at CIS [12] and, lately, at symmetry-adapted cluster-configuration interaction SAC-CI level [13]. From the evolution of the computed transition energies (, upper panel) and DCT values (mid panel), reported in Fig. 3, it is clear that for P1 all methods provide a consistent picture, corresponding to a charge transfer from the donor group (D = NH2 ) to the acceptor (A = NO2 ). Indeed in this case a CT transition with substantial overlap between the centroids of charge representing the zones of increase and decrease of electron density upon excitation is predicted since the computed value of DCT is very close to the H value and consequently the t index, defining the through space character (Fig. 3, lower panel), is computed to be very small by all methods. As a consequence it is not surprising that for P1 all methods provide reliable results of the energetics associated with this transition (i.e. computed transition energy close to the experimental value). On the contrary, starting from P2, some clear differences between the results obtained at PBE and PBE0 level from one side and LC-PBE from the other side can easily be noticed. In particular if both PBE and PBE0 always predict a Donor to Acceptor transition with through space character, as demonstrated by the associated ˚ respectively for large values of DCT and t (of 5.0 A˚ and 2.8–2.9 A, ˚ implying P2), LC-PBE predicts a much shorter DCT value (of 3.7 A) a more active role as donor of the phenyl bridge connected to the ˚ acceptor group and a smaller through space character (t = 0.5 A). Actually, starting from P3 the difference gets even more pronounced since PBE and PBE0 still continue to predict a donor to acceptor transition with increasing through space character while LC-PBE predicts essentially a bridge to acceptor transition with constant CT character (that is constant DCT ) and no relevant through ˚ As a consequence, due to the space character (t always below 0.5 A). well-known incorrect asymptotic behavior of both PBE and PBE0, the computed transition energies associated with the through space transition are greatly underestimated as confirmed by the large discrepancy obtained starting from P2 at these levels of theory. On the other hand, from an energetic point of view LC-PBE (possessing the correct 1/r asymptotic behavior) provides a consistent evolution of the computed transition energies with respect to the experimental data. On this basis, and also supported by the results obtained at CIS level which are completely parallel with the LC-PBE results both in terms of energetic and DCT evolution one would conclude that for transition with relevant through space ˚ GGAs or character, here quantified by a t value larger than 1.6 A, global hybrids are not reliable for the prediction of both transition energy and the nature of CT transitions. Indeed, more recent reference calculations performed at SAC-CI level in the case of the P1, P2 and P3 systems [13] showed that, interestingly, the nature of the transition (that is, a donor to acceptor excitation) seems to be correctly described at PBE0 level as demonstrated by the equivalent DCT and t values computed with this functional and at SAC-CI level, and that only the associated

O 2N

6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5

PBE PBE0 LCPBE CIS SACCI

1

2

3

4

n Fig. 3. Transition wavelengths (␭ in nm), DCT and t values (in Å) computed at different level of theory using the 6-31+G(d) basis on PBE0/6-31+G(d) optimized structures for the Pn systems (n = 1–4). Solvent (EtOH) is included using the Polarizable Continuum Model (C-PCM). Values from Refs. [12,13].

C. Adamo et al. / Coordination Chemistry Reviews 304–305 (2015) 166–178

A

171

D

1

A

D

2 A D

3 A=DCI NC

A

D

N

4 NC

D= H, OMe, NMe2

N A D

5

A

D

6 Fig. 4. Set of push–pull molecules considered in Ref. [18].

transition energies are largely underestimated at TD-DFT level. This point is particularly relevant since it somehow questions the use of range-separated hybrids, even though they predict the correct transition energy evolution, for the description of the nature of the excited states. The general conclusions on functional performance in the description of CT excitation nature and energetics were further confirmed by a successive study [18] which considered a larger panel of push–pull systems (depicted in Fig. 4) and of functionals belonging to the GGA, global hybrid and range separated hybrid family (namely PBE, BLYP, PBE0, B3LYP, BHandH, M06, M06-HF, CAM-B3LYP, LC-PBE) together with reference CIS calculations. For this set of 18 molecules, with the same acceptor unit, namely the 4,5-dicyanoimidazole (DCI), three different types of donor (H, OMe and NMe2 ) and six types of bridge (each having a different degree of conjugation, 1–6, Fig. 4), the range separated hybrids and global hybrid with high percentage of HF exchange actually provide a more localized description of the transition with the bridge acting as a donor while global hybrids provide a larger CT distance. In particular, the analysis of the LCT charge transfer efficiency parameter, defined as the ratio between the physical distance between the donor and the acceptor groups (D0 ) and the computed DCT value, clearly shows that the range separated hybrids provide a much more localized picture (i.e. an higher LCT value) of the transition with respect to global hybrids (as depicted in Fig. 5). Indeed all the functionals predict that a complete and ideal transfer from the donor to the acceptor is practically never obtained in presence of a bridge as LCT is almost always predicted to be larger than 1. The strong effect that functionals may have on properties of push–pull systems is also clearly highlighted by the computed nonlinear optical properties of such compounds [67]. Indeed, a strong dependence of these properties on the exchange correlation functional used is expected and a recent work of Sun et al. [67] has shown that a non-empirical tuning of hybrid functional with range separated exchange can yield a physically sound chemical picture correcting the overdelocalization provided by standard functionals.

Fig. 5. Computed LCT values (LCT = D0 /DCT ) at CAM-B3LYP (upper panel) and B3LYP (lover panel level). Refer to Ref. [18] for further computational details and to Fig. 4 for the representation of the molecular systems considered.

From a more chemical point of view, the work [18] on push–pull systems also demonstrate that there is a clear impact of the bridge nature on the CT efficiency in term of spatial extent. Indeed it is observed, for instance, how a biphenyl bridge can play the role of a primary donor more than that of a simple bridge resulting in reduced CT distance. This kind of information can thus be used for the design of new molecular devices with enhanced CT properties.

4. Application of DCT to describe and quantify the charge transfer at the excited state in molecular compounds After its proposal in 2011, the DCT index was indeed mainly applied as a measure (qualitative and quantitative) of the charge transfer extent associated with electronic excitations in molecular systems. Using such a tool, the nature of the excited states of different molecular systems (both fully organic and containing metals) was analyzed in order to understand, or enhance their excited state properties. Indeed long range charge transfer (CT) up to Charge Separation (CS) at the excited state is a target for many applications of molecular compounds ranging from artificial photosynthesis, solar cells and non linear optical materials to Near InfraRed (NIR) dyes for bioimaging and sensising and optoelectronics. Not surprisingly, many combined experimental and theoretical works where the DCT index has been used in these respects can be found in the design of new dyes for solar cell applications [17–37], and in the rationalization of the properties of NIR chromophores [56–60], NLO molecular material [43–47] and fluorescent molecules [38–42].

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Me2N

NO2

NO2 Me2N

n

Me2N

S

NO2

n

n-1

Me2N

Se

n

NO2

Me2N

O

n

NO2

S

Me2N

N H

n

Me2N

NO2

NO2

S

n

N Me2N

NO2

Me2N

NO2

n

n

Me2N

NO2

n-1 Fig. 6. Schematic representation of the different oligomeric bridges considered in Ref. [17] (n = 1, 10). The oligomeric thiophene bridge is highlighted in green.

To achieve a controlled and directional CT or CS at the excited state, for instance, the synthetic strategy mainly relies on the use of push–pull systems (eventually rod-like) essentially composed of a spacer () connecting an electron donor (D) to an electron acceptor (A) with the aim of promoting, upon irradiation, an electron transfer from the donor to the acceptor moiety while achieving a formally [D+ ––A− ]* excited state. Nonetheless, experimentally optimizing the CT phenomena in such systems is far from trivial since this process relies on a very subtle interplay of the electronic and geometrical coupling between the D and A units through the bridge as well as in the modulation of the D and the A strength. In this respect the DCT index enables the measurement of the CT extent for molecules belonging to the same family as a function of the Bridge, Donor or Acceptor type (and bridge length) and it provides a powerful tool for the rational design of new molecules. In a recent paper [17], the DCT index was first applied to characterize the role of the bridge in D––A molecules containing the same donor ( NMe2 ) and the same acceptor group ( NO2 ) but 10 different types of bridge schematically depicted in Fig. 6 in conjunction with the CAM-B3LYP functional, a level of theory that grants for the reliability of the excited state energy. From the analysis of the DCT and qCT it turns out that there is a typical saturation length of the bridge (around 4–5 oligomers) leading to the maximal DCT value and a corresponding asymptotic transition wavelength. Longer bridges do not, in fact, improve the CT distance and yield practically the same excitation wavelength. Analysis of the DCT values and of the density difference plots enables one to clearly identify that the longer bridges actually act as a donor, the transition thus becomes shorter range corresponding to a bridge-to-acceptor excitation. From this analysis, the pentathiophene appeared as the best compromise and it was used to screen

between the different donor and acceptor groups. Using this bridge it was possible to see the effect of varying the donor group at fixed acceptor ( NO2 ). Indeed, no very sizable difference between the donors of different strength (namely NMe2 , NH2 NHMe, NPh2 , OMe, 4-Ph(NPh2 ) and 4-Ph(N(PhOMe)2 ) was noticed since, as discussed above, the bridge plays the role of the primary donor, while a larger modulation of the CT length was found by changing the acceptor at fixed (NMe2 ) donor. In particular, it was possible to clearly observe that the NO2 and the multicyano acceptors are one of the strongest, yielding a DCT of roughly 5 A˚ (or larger) while providing a large spectral diversity, with associated predicted absorption wavelength from 458 nm (for the NO2 group) to 527 nm (for the multicyano) [17]. In the same context, i.e. charge transfer distance in push–pull dyes for DSSCs applications, the effects of the change in acceptor groups was studied by Zhang and collaborators [19] while Li and collaborators [33] and He and collaborators [37] analyzed the role of different spacers. Indeed, and not very surprisingly, most of the times the DCT index was applied (in a more or less systematic way) to quantify the strength and the sign of CT in dyes used in solar cells applications and to compare similar classes of dyes obtained by functionalization of a similar skeleton either fully organic [17–29] or made up by a transition metal complex [30–37]. While the majority of the examples presented up to now deals with organic molecules, it should be stressed that the density difference maps () and the associated DCT indexes were also applied to metal containing complexes.  and DCT indexes were indeed first used to obtain a more straightforward description of the excited state nature in transition metal complexes allowing one to condense the information contained in the description of electronic

C. Adamo et al. / Coordination Chemistry Reviews 304–305 (2015) 166–178

N

173

N+

O

χ1

COO-

RhodB

χ2 Fig. 7. Computed and experimentally estimated DCT values for two ruthenium complexes from Ref. [44].

transitions computed at TD-DFT level. Indeed, in transition metal complexes, electronic transitions are generally resulting from a linear combination of several mono-electronic transitions making their interpretation and assignment often difficult. The use of density-based indexes allows one to overcome this problem to define the character of an electronic excitation (ex. Ligand Centered, MLCT, Locally Excited or Inter Ligand for instance). For these reasons, these indexes were recently applied, for instance, to analyze all the electronic transitions occurring in the visible range for a series of ruthenium complexes and zinc porphyrins commonly used in Dye Sensitized Solar Cells [31,34,36,68]. One very interesting use of the DCT indexes on complexes comes from the work of Coe et al. [44] who determined experimentally, using Stark spectroscopic measurements, the values of DCT and  in a series of ruthenium complexes and compared it to values computed from TD-DFT calculations performed at global hybrid level using the B3LYP functional. The computed values of DCT were actually in fairly good agreement with the experiment as shown in Fig. 7. Using the same experimental procedure, Coe et al. were able to estimate experimentally the DCT values of a series of ferrocene molecules. These results further highlighted that, if a reasonably accurate computational protocol is used to determine ground and excited state density, the DCT index has a clear physical meaning [69]. Indeed it is worth stressing that if the majority of the push–pull molecules analyzed with the DCT index are designed for DSSC application, the DCT index was also used to characterize the nature of the transition of push–pull systems designed for NLO applications [43–47] or fluorescent probes and luminescent families of molecules [38–42,48–60] with the aim of giving a qualitative interpretation of their photophysical behavior. 5. Density-based indexes for the description of excited state evolution and reactivity. Beside a description of the nature of vertical excited states, as described in the previous section, the DCT index has also been applied to characterize excited states evolution [51,52,54,55]. Indeed, several photophysical processes involve the crossing of excited states whose characteristics may be completely different. In this respect, a few recent studies making use of the DCT index specifically addressed the description of the excited state evolution in dyes belonging to the Rhodamine family [51,52]. Indeed, different dyes belonging to the same Rhodamine family present different fluorescence quantum yields and, due to their molecule dependent very low intersystem crossing rate. The most plausible explanation is related to the presence of an internal interconversion responsible for the non-radiative decay. Beside other proposals, it was recently shown how the interconversion between a bright (Locally Excited, LE) and a dark state (Charge Transfer, CT)

N

N+

O

COO-

5TMR

COOH

N

N+

O

COO-

Rhod101

Fig. 8. Schematic drawings of different Rhodamine molecules considered in Ref. [52].

can be considered as the mechanism responsible for the fluorescence quenching [52]. In this respect, three Rhodamine derivatives (namely RhodB, 5TMR and Rhod101, Fig. 8), experimentally characterized by markedly different fluorescence quantum yields, were recently theoretically investigated by the means of TD-DFT at global hybrid level [70] and their overall photophysical behavior was rationalized using density-based indexes [52]. First of all, the three molecules show different structural features at the ground state: RhodB and 5TMR have a quite symmetric structure (characterized by a 1 close to 90◦ and a ␹2 close to 0◦ ) while Rhod101 is not symmetric, the carboxyl group being tilted of about 30–40 degrees with respect to the phenyl ring (this latter being far from orthogonality with respect to the xanthene plane). Conversely, the lowest lying singlet excited state (S1 ), computed as vertical excitation from the ground state minimum, for each of these three Rhodamines always corresponds to a LE state associated with a - excitation localized on the xanthene moiety. While Rhodamines possessing a symmetric ground state

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3.0

Table 1 First two (S1 and S2 ) vertical excited state energies (in eV), oscillator strength and character (LE, locally excited; CT, charge transfer) computed for RhodB, 5TMR and Rhod101 from their ground state structures (from Ref. [52]). S2

2.66 (0.97) LE 2.66 (0.91) LE 2.35 (1.26) LE

2.96 (0.13) CT 2.98 (0.13) CT 2.86 (0.02) LE

2.5

2.0

DCT

RhodB 5TMR Rhod101

S1

1.5

1.0

0.5 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.6

0.7

0.8

0.9

1.0

cLSP

4.0 3.5 3.0 2.5

E (eV)

geometrical structure (i.e. RhodB and 5TMR) are characterized by a second–dark–excited state (S2 ) of significant charge transfer character (from the carboxyphenyl to the xanthenes moiety), this is not the case for the asymmetric Rhod101 system displaying a second excited state still of LE character (as reported in Table 1). Relaxation of the first (LE) excited state in the Franck–Condon region thus always yield to a stable bright LE mimimum, symmetric in the case of RhodB (2.41 eV) and 5TMR (2.40 eV) and asymmetric in the case of Rhod101 (2.25 eV). Interestingly, in the case of Rhod101 no stable symmetric structure can be located on both the ground and excited state PES. Nonetheless, in the case of RhodB and 5TMR it is also possible to optimize the first excited state imposing a non symmetric starting structure. In this case a new minimum (lower in energy with respect to the symmetric ones) is found on the S1 PES of both RhodB (1.46 eV) and 5TMR (1.95 eV) corresponding to a dark state responsible for their observed low quantum yield. Both these minima are characterized by a 1 of roughly 60◦ and a 2 around 20◦ . In order to characterize the interconversion process taking place at the excited state and responsible for the quenching of fluorescence in the case of RhodB the energies and DCT values associated with the S1 and S2 excited states computed on a linear synchronous path connecting the two S1 minima were thus calculated. The results obtained are depicted in Fig. 9. First of all it is easily noticed that the evolution of the S1 and S2 energies nicely parallel that of the associated DCT index. As previously discussed, around the symmetric minimum in the Franck–Condon region (cLSP = 0), the first excited state (S1 ) corresponds to a bright LE state associated with a low DCT while the second excited state (S2 ) lies relatively higher in energy. This state is characterized by a much larger DCT value and actually corresponds to a dark CT state. As soon as the molecule starts to convert toward the asymmetric minimum (that is moves toward the cLSP = 1 region), the difference in energy between the S1 and S2 states decreases and the character of the states mix up, the S1 state gaining in CT character and the S2 state becoming more local in character, as demonstrated by the evolution of the DCT index associated with the two states in the region 0.2 < cLSP < 0.4. After cLSP = 0.4 the nature of the two state is inverted with respect to the symmetric minimum as measured by the corresponding DCT . From a mechanistic point of view, the inversion of the two excited states measured by the DCT is related to the inversion of donor group (from the xanthene and the carboxylate group) along the linear synchronous transit coordinate from the symmetric to the asymmetric minimum. From an orbital point of view, the quenching mechanism is triggered by the destabilization of orbital centered on the carboxylate group (i.e. the HOMO-1 in the case of the symmetric minimum). This quenching mechanism is thus active only for rhodamines showing a substantial contribution from the carboxylate group to the frontier orbitals or, in other words the lowest lying CT states. This is actually not the case for all asymmetric rhodamines since the carboxyl group is geometrically decoupled from the rest of the molecule and always contributes to more internal orbitals. Overall, it is interesting to note how the combined use of TD-DFT and DCT , enables us to clearly follow and rationalize the different

2.0 1.5 1.0 0.5 0.0 0.0

0.1

0.2

0.3

0.4

0.5

cLSP Fig. 9. Upper panel: evolution of the DCT index going from the symmetrical (cLSP = 0) to the asymmetrical (cLSP = 1) RhodB S1 structures computed for the first and the second excited states. Lower panel: Energies associated with the first and second excited state of RhodB along the linear synchronous transit path (from Ref. [52]).

H2N

NH2+

O

COO-

Rhod110 Fig. 10. Schematic structure of Rhod110.

photophysical behavior of molecules that are apparently very similar (all rhodamines) yet show very different quantum yields. Focusing on one asymmetric Rhod110 molecule (Fig. 10), it was also possible to show how the use of DCT and DCT,react indexes allows one to identify the driving force of the reorganization process taking place at the excited state [51]. Indeed, if this molecule shows an

C. Adamo et al. / Coordination Chemistry Reviews 304–305 (2015) 166–178

175

Fig. 13. Schematic representation of two ES proton transfer mechanisms investigated: intermolecular ESPT occurring between CouOH and 1-MeId (left, from Ref. [54]) and intramolecular ESPT occurring in HBT (right, from Ref. [55]).

Fig. 11. GS (S0 , black line) and ES (S1 , red line) Rhod110 energetic profiles (in eV) computed between ground state minimum energy structure (cLSP = 0) and excited state minimum energy structure (cLSP = 1). Associated difference density plots () computed for the minimal energy structures are also reported (positive and negative variations of the density are represented in dark and light blue, respectively).

Fig. 12. Evolution of DCT,react (in Å) along the linear synchronous path coordinate for Rhod101 (from Ref. [51]).

asymmetric structure at both the ground and the excited state, as in the case of Rhod101 previously discussed, a larger decoupling of the phenyl ring is computed at the excited state (with an increase of roughly 10◦ with respect to the ground state structure). This geometrical decoupling allows the increase of the partial Intramolecular Charge Transfer (ICT) character associated with this electronic transition, although the character of the transition itself is not substantially modified as can be clearly seen from the density difference plots associated with the vertical (absorption) and relaxed (fluorescence) S1 states reported in Fig. 11. The structural relaxation on the S1 PES allows the slight increase in the ICT character, as described by the increase of the DCT index along a linear synchronous path connecting the ground state ˚ to the excited state one (cLSP = 1, minima (cLSP = 0, DCT = 1.44 A) ˚ Nonetheless, the driving force leading to the stabilizaDCT = 2.12 A). tion of the excited state along this path can be clearly related to the minimization of the charge separation along the path as described by the decrease of the DCT,react (Fig. 12). The potential of density-based indexes when describing excited state reactivity was recently more deeply elucidated by considering

a very specific class of ES reactions that is ES Proton Transfer (ESPT) occurring either at the intermolecular or intramolecular level. Indeed, beside experimentally playing a central role in many biological and energy conversion processes, ESPT reactions are actually very challenging from a computational point of view since the quantitative modeling of such processes implies a correct description of an extremely flat excited state PES and very fine control of the environmental effects (such as solvent) which tune the reactivity. Generally speaking, ESPT reactions imply that, upon electronic excitation induced by the absorption of light, the acidity and basicity of the proton donor (ex. OH) and acceptor (ex. a N or S atoms) groups, respectively, are enhanced as a consequence of the redistribution of electronic density across the molecule. In such a way, proton transfer from the proton donor to the acceptor groups can take place at the ES while being thermodynamically impeded at the ground state. In such a context it is somewhat clear that indexes measuring the electron density reorganization upon excitation can be extremely useful in the description and rationalization of these processes. Recently two different types of ESPT reactions, already thoroughly experimentally investigated, were rationalized using density-based indexes: (1) an intermolecular ESPT [54], occurring between the 7-hydroxy-4(trifluoromethyl) coumarin (CouOH) and 1methylimidazole (1-MeId, left panel of Fig. 12) and (2) an intramolecular ESPT [55] taking place within the 2-(2hydroxyphenyl) benzoxazole (HBT, right panel of Fig. 13). In the latter case the ESPT reaction basically corresponds to a tautomerization converting between the enolic and ketonic form of HBT. In order to study and characterize the ES reaction, one requires at least two degrees of freedom, namely variation in the donor atom (here oxygen) to proton distance as well as the donor to acceptor atom internuclear distance (here represented by the O N distance) since a shortening of the O N distance is concomitant with the minimal energy path at the ES. Indeed, this is what is observed for both reactions since the minimal energy path goes through a contraction of the ON distance followed by an elongation of the OH distance, as can be easily deduced by inspection of the upper panels of Fig. 14. More interestingly it appears that for both reactions the computed excited state PES can be mapped by the evolution of the DCT index. In particular, it is predicted that, as the proton is transferred, the DCT index increases up to maximum value, corresponding to the Transition State for the transfer, before decaying to the final value corresponding to the CT distance computed for the products. It is interesting to note that the DCT value does not decay significantly after proton transfer in the case of the HBT tautomerization. This is due to the intramolecular nature of the reaction, in which the proton is somewhat held in place after the reaction has completed. In the other case previously cited, that is for the CouOH+1-MeId reaction, the difference between the DCT values computed between

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Fig. 14. Upper panel: Two dimensional excited state potential energy surface (PES) computed in toluene for the CouOH+1MeId ESPT reaction (total energy in a.u.; left) and for the HBT tautomerization (relative energy in kcal/mol; right). Lower panel: Two dimensional DCT surfaces computed in toluene. for the CouOH+1Me-Im ESPT reaction (left) and for the HBT tautomerization (right). DCT , OH and ON distances are in Å. Black squares indicate the values of the OH distance (in Å) along the minimal energy path.

the reactant and the product of the ESPT is larger due the possibility of observing a more dramatic decrease in the charge separation because of intermolecular nature of the event and thus of possibly more pronounced structural reorganization. As previously mentioned, the DCT,react index, combining the effect of both nuclear and electronic reorganization occurring during an ES reaction, can be very suitable for the definition of adapted ES reaction coordinates. Indeed, computing the DCT,react for the PES calculated in the case of the CouOH+1-MeId (as depicted in Fig. 15)

Fig. 15. Evolution of the DCT,react (in Å) as a function of the ON and OH distances computed for the intermolecular ESPT reaction between CouOH and 1-MeId in toluene solution. Black squares indicate the values of the OH distance (in Å) along the minimal energy path.

led to the possibility of the clear identification, in the meaningful region of small displacements around the ground state structure considered as a ground state reference for the computation of the DCT,react , of the most favorable paths for the excited state evolution. Indeed, it is evident that the DCT,react decreases going from the reagents to the products and that it is possible to locate the minimum energy path of the intermolecular ESPT reaction by following the steepest variation pathway of the index. Therefore, from the results obtained we can see how the combined use of DCT and DCT,react allows one to follow and define the most relevant path of ES evolution in the case of ESPT reactions, as well as to rationalize the driving force of the ES reactions. Clearly, in the more general case of intermolecular ESPT reactions, the presence of protic solvents or of high concentration of base molecules can significantly increase the complexity of the model system to be considered. Nonetheless, even more complex phenomena related to the ESPT can nowadays be studied, as demonstrated by the recent investigation of the mechanism of the photo-tautomerism [71] occurring at high concentration of base molecules in the case of the CouOH+1MeId systems, schematically depicted in Fig. 16. From a mechanistic point of view, the results obtained in this study indeed confirm the previous findings concerning the role of the coumarin photoacidity which is actually the overall driving force of the entire shuttling process responsible for both the first proton transfer event and of the enhancement of the subsequent base to base proton hopping processes.

C. Adamo et al. / Coordination Chemistry Reviews 304–305 (2015) 166–178

CF3

O

CF3

O

O H

Me N

N Me

N

N Me

O

hν

N

Me

N

N N Me

O

N

O

+ H N

Me

N

N

Me

-O

hν N N Me

N Me

N N

O

O

N

Me

N

N

Me

CF3

*

+ H N

Me

N

O N

* O-

O

O H+ N

CF3

O

N

O-

O

Me

CF3

-

CF3

*

(a) N

177

+ N H

N

O-

O

N H+

N

*

Me Me N

Me N

N

N

Me

N

N

Me

Fig. 16. Schematic process modeled to rationalize the phototautomerization of CouOH at high concentration on 1 MeId in toluene solution as described in Ref. [70].

6. Conclusions From their inception, density-based indexes have been extensively used to describe (from a qualitative and quantitative point of view) the nature of the electronic reorganization occurring upon an electronic excitation. Due to their very simple and chemically sound definition, these indexes convey the simple chemical concept of charge transfer so that they are of great help in the description of the excited state nature and in the definition of the CT. For these reasons they have been used to rationalize and design (from a computational point of view) new molecules, especially when CS or CT is an important target criteria, such as in the case of dyes for photovoltaic applications, fluorescent probes or materials for non-linear optics. Indeed, despite their limits, these indexes can be computed at practically no extra cost and provide a simple measure of TD-DFT accuracy and while affording a cheap comparison between the results obtained at different level of theory. More recent works have also shown their potential use in the exploration of potential energy surfaces at the excited state. Indeed, these indexes enable to follow a selected excited state among closely lying states and to identify maxima and minima over a single ES-PES. Work is currently in progress in this respect in order to better explore their potential in conjunction with dynamic approaches or to study the fate of an excited state after the first irradiation event.

[4] [5] [6]

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

Acknowledgments

[25]

I.C. acknowledges the members of COST Action CM12002 for stimulating discussions. M. E. acknowledges the support by a Grant-in-Aid for Scientific Research from JSPS and Nanotechnology Platform Program of MEXT of Japan.

[26]

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