Charge-displacement analysis for excited states Enrico Ronca, Mariachiara Pastore, Leonardo Belpassi, Filippo De Angelis, Celestino Angeli, Renzo Cimiraglia, and Francesco Tarantelli Citation: The Journal of Chemical Physics 140, 054110 (2014); doi: 10.1063/1.4863411 View online: http://dx.doi.org/10.1063/1.4863411 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ensemble density functional theory method correctly describes bond dissociation, excited state electron transfer, and double excitations J. Chem. Phys. 142, 184104 (2015); 10.1063/1.4919773 Charge-displacement analysis via natural orbitals for chemical valence: Charge transfer effects in coordination chemistry J. Chem. Phys. 142, 084112 (2015); 10.1063/1.4908537 Charge transfer in strongly correlated systems: An exact diagonalization approach to model Hamiltonians J. Chem. Phys. 140, 134101 (2014); 10.1063/1.4869520 Surface hopping dynamics using a locally diabatic formalism: Charge transfer in the ethylene dimer cation and excited state dynamics in the 2-pyridone dimer J. Chem. Phys. 137, 22A514 (2012); 10.1063/1.4738960 Response calculations based on an independent particle system with the exact one-particle density matrix: Excitation energies J. Chem. Phys. 136, 094104 (2012); 10.1063/1.3687344

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THE JOURNAL OF CHEMICAL PHYSICS 140, 054110 (2014)

Charge-displacement analysis for excited states Enrico Ronca,1,2,a) Mariachiara Pastore,1,b) Leonardo Belpassi,1 Filippo De Angelis,1 Celestino Angeli,3 Renzo Cimiraglia,3 and Francesco Tarantelli1,2,c) 1

Istituto CNR di Scienze e Tecnologie Molecolari, via Elce di Sotto 8, I-06123 Perugia, Italy Dipartimento di Chimica, Biologia e Biotecnologie, Università degli Studi di Perugia, via Elce di Sotto 8, I-06123 Perugia, Italy 3 Dipartimento di Scienze Chimiche e Farmaceutiche, Università degli Studi di Ferrara, via Borsari 46, I-44100 Ferrara, Italy 2

(Received 18 November 2013; accepted 15 January 2014; published online 5 February 2014) We extend the Charge-Displacement (CD) analysis, already successfully employed to describe the nature of intermolecular interactions [L. Belpassi et al., J. Am. Chem. Soc. 132, 13046 (2010)] and various types of controversial chemical bonds [L. Belpassi et al., J. Am. Chem. Soc. 130, 1048 (2008); N. Salvi et al., Chem. Eur. J. 16, 7231 (2010)], to study the charge fluxes accompanying electron excitations, and in particular the all-important charge-transfer (CT) phenomena. We demonstrate the usefulness of the new approach through applications to exemplary excitations in a series of molecules, encompassing various typical situations from valence, to Rydberg, to CT excitations. The CD functions defined along various spatial directions provide a detailed and insightful quantitative picture of the electron displacements taking place. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863411] I. INTRODUCTION

In the past decades a growing attention has been devoted to the comprehension of charge transfer (CT) processes occurring in excited molecular systems, due the crucial role they play in many fields of the physical, chemical, and biological research.1, 2 In nature, for instance, the most important constituents of life, including nucleic acids,3–6 proteins,7, 8 and light harvesting chromophores (photosynthetic systems),9 operate thanks to efficient charge exchanges between chemical species.10–12 Molecular systems with intense and long-lived CT excitations are also widely exploited in the construction of technological devices, mainly in the field of renewable energies, where CT excitations initiate solar energy conversion into electricity (Hybrid13–15 and Organic Photovoltaic16–18 ) or trigger reactions suitable for H2 production (water splitting cells).9, 19 Also, CT excitations find significant applications both in the field of optoelectronic devices, for the production of light emitting screens (OLED),20, 21 and of nonlinear optics, where their second-order optical properties are often exploited in material science.22, 23 From the experimental side, several techniques have been used to investigate CT in molecular systems,24–32 although, only few of them are able to quantitatively estimate the amount of transferred charge, particularly when excited states are concerned.33 Also, a number of theoretical models, principally based on orbital localization techniques,34–40 have been developed for the evaluation of CT in the ground state, but few approaches allow the accurate characterization of charge displacements (CDs) occurring during an electronic excitation. a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected]

0021-9606/2014/140(5)/054110/13/$30.00

Generally, the standard theoretical approach used to describe the charge rearrangements following a transition, is based on the analysis of the molecular orbitals (MOs) involved in the excitation, under the approximation that both the ground and electronic excited states are well described within an orbital model.14, 41–43 This strategy, if on the one hand provides a simple chemical picture of the excitation process, on the other hand clearly fails when the electronic states present a marked multideterminantal character or when a marked modification of the occupied orbitals not involved in the excitation is observed; furthermore, it is generally difficult and unreliable to obtain an estimate of the actual amount of charge moving during the excitation. Usually, to gain quantitative information on the electronic rearrangement, one resorts to population analyses, which however, as is widely known for the ground state, are typically model-dependent and require a tedious analysis of both the ground and excited state wave functions. An alternative approach to obtain information about an electronic transition is based on the analysis of the single particle transition density.44–46 This couples the electronic ground state with the excited state under consideration, providing important information about the characteristics of the transition. As a matter of fact, however, the deconvolution of the transition density is not a trivial procedure, in particular for large-size molecules. In this context, an easier strategy is represented by the use of the natural transition orbitals (NTOs), which permit to express an electronic excitation as a single particle-hole pair;47–49 the transition density analysis, however, cannot be used to extract quantitative information on the electron movements occurring during the excitation. A completely different methodology, often used to identify the nature of an electronic transition, is the analysis of the difference density between ground and excited states.50–52

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Despite its possibly intricate structure, this function has been successfully applied to characterize valence, Rydberg, and CT n → π * and π → π * states in a variety of organic molecules. An appealing and physically meaningful extension of this model is based on the analysis of the so called Attachment (A) and Detachment (D) densities,53–58 defined as the sum of the eigenvectors obtained from the diagonalization of the difference density matrix corresponding to negative and positive eigenvalues, respectively. In this formulation the detachment density can be seen as a representation of a hole, while the attachment one can be associated with a particle, allowing the simple description of the electronic excitations as a D → A transition. Although this approach provides a clear understanding of the nature of the excitation, it cannot be used to estimate the extent of the charge displacements. The problem of defining a quantitative parameter expressing the degree of CT has also been addressed within the Time-Dependent Density Functional Theory (TDDFT) framework, with the aim of setting up a diagnostic tool for evaluating the expected accuracy of the computed excitation energies and gaining insight into the nature of the excited states.59–61 Finally, other strategies commonly used to characterize a transition are based on the study of some observable properties: the analysis of the R2 expectation values, for instance, can give information on the Rydberg and CT nature of the excited states. In recent years, we have proposed a novel approach to evaluate CT occurring during the formation of a bond, known as the charge displacement analysis.62–64 This method, based on a partial integration of the electron density difference between a bound system and its separated fragments, provides information about the spatial extent of the charge rearrangements and permits an accurate evaluation of the amount of electrons that have been exchanged by the fragments during bond formation. The application of this model led to significant progress in the comprehension of the bond in various systems, such as weakly bound water adducts,63, 65, 66 organometallic complexes,67 and atoms interacting with metal surfaces.68 Recently, this model has been also used to investigate the electronic properties of the interface between organic dyes and a semiconductor (TiO2 ) surface in dye-sensitized solar devices.69 In the present paper, we extend the CD approach to the characterization of electronically excited states, obtaining both qualitative and quantitative information about the spatial extent of the electron movements and CT. After a brief summary of the CD analysis and its straightforward extension to study electronic excitations, we shall first exemplify the kind of information and insight which such excited-state charge displacement (ESCD) method provides, by using it for the characterization of some selected excited states of the H2 molecule, for which very accurate full configuration interaction (FCI) calculations can be performed. We shall then present case studies of excited states of different nature, such as valence, ionic, Rydberg, and CT states, in small benchmark organic molecules, using highly correlated Multireference Perturbation Theory methods (NEVPT) developed by some of us70–75 and already successfully applied to the description of the spectroscopic properties of both organic and inorganic molecules.76–82

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II. EXCITED-STATE CHARGE DISPLACEMENT

Until now, CD analysis has been applied to the study of the electron density changes taking place upon formation of bonds between atomic/molecular systems. In this case the electron density change ρ has been defined as the density difference between the bound complex and the isolated noninteracting partners. We have then focused our attention on the CD function defined by  +∞  +∞  z  dz ρ(x, y, z ) dx, dy. (1) q(z) = −∞

−∞

−∞

q(z) measures exactly, at each point along a chosen z axis, the electron charge that, upon bond formation, is transferred from the right to the left side of the perpendicular plane through z;62 a negative value thus corresponds to electron flow from left to right. In this way, the method gives a clear picture of the direction and extent of electron transfer (ET) over the whole molecular region without needing any particular model of charge decomposition. From Eq. (1) one notices that the CD approach can be meaningfully applied whenever it is possible to define a ρ between an “initial” and a “final” state as a consequence of some process. Therefore, the extension of the method to the description of the electron density changes occurring during an electronic excitation is straightforward, as one only needs to redefine ρ as the difference between the excited and ground state electron densities. It should be mentioned here that, while the evaluation of the ground state density can be considered an easy step at any level of calculation, this might not be the case for the excited state, since all the theoretical methods based on the response do not allow a direct evaluation of the excited state electron density. The detailed analysis of this issue is, however, outside the scope of this paper, where we shall make use of accurate multireference methods which provide the wave function and consequently the charge density of both the ground and excited states. III. COMPUTATIONAL DETAILS

The calculations of the three lowest-energy transitions of the H2 molecule have been performed at the full configuration interaction level using the aug-cc-pVQZ basis set.83 In order to better describe the Rydberg molecular orbitals, the above basis set was augmented with molecule-centered uncontracted 8s8p8d diffuse functions (in this case the center has been placed in the middle of the H–H bond), whose exponents were optimized as described by Kaufmann et al.84 The H2 calculations have been performed by using the MOLPRO 2010 software package.85 We have then carried out the ESCD analysis of benchmark organic molecules: formaldehyde, ethylene, pyrrole, and a prototype mixed–valence compound. For these systems accurate electron densities and excitation energies have been obtained by performing zero-order Complete/Restricted Active Space (CAS/RAS)86, 87 self-consistent field (SCF) calculations followed by NEVPT calculations to take into account the dynamic correlation. The NEVPT approach has been applied both in its strongly contracted (SC) and partially

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contracted (PC) variants,70–74 both to second and, in some cases, to third order of perturbation. The calculation of the electron density has been carried out at the complete active space self-consistent field (CASSCF) level and also at the second order of perturbation theory in the NEVPT approach, adopting a code developed by the authors of the SC– NEVPT3 implementation.74 As will appear in the following, the correction brought to the CASSCF density by the second order approach is very small, warranting the neglect of the perturbation correction. Therefore, for these systems, the ESCD analysis has been performed using the unperturbed excited and ground state CASSCF/restricted active space selfconsistent field (RASSCF) electron densities. The basis sets used, geometries, and active space composition are separately reported below for each system investigated. IV. RESULTS AND DISCUSSIONS A. A first preliminary case: the hydrogen molecule

We chose the H2 system as preliminary test case to apply the ESCD analysis for two basic reasons: its small size, which permits an accurate FCI treatment, and the well characterized nature of its lowest transitions, amply documented in the literature.88–99 We have considered the two lowest-energy singlet-singlet excitations (X 1  g → 1 B 1 u+ 2pσ and X 1  g → 1 C1 u 2pπ ) and the first singlet-triplet transition (X 1  g → 3 B3 u+ 2pσ ). The FCI treatment of both ground and excited states provides excitation energies which match exactly the experimental values (see Table I) and equally accurate electron densities should be expected. The 1 B excited state can be qualitatively described as the excitation of one electron from the double occupied σ g orbital to the antibonding σ u orbital. The shape of these molecular orbitals has been accurately characterized by Rothenberg and Davidson.95 They thus motivated the increase in bond length accompanying the excitation with the pure antibonding character of σ u , denominated 2pσ u . Indeed, this orbital shows a node in the middle of the H–H bond and two evident lobes extending outside the intermolecular region along the bond axis. Contour plots of the FCI electron density change accompanying the excitation are displayed in Figure 1 from two different viewpoints, where z is the internuclear axis and x lies perpendicular to it. The corresponding q(z) and q(x) CD functions are also shown. Figure 1 gives a clear idea of the restructuring of the charge upon excitation. In particular, in the top left panel of TABLE I. Calculated and experimental vertical (ω) excitation energies and differences between the excited and ground states potential energy minima (Te ) of H2 . Excitation X → 1B X → 1C X → 3B a

ωFCI (eV)

TFCI (eV) e

12.74 13.20 10.60

11.37 12.40

References 101–103. References 103–106. c Reference 100. b

ωexp (eV)

exp

Te

(eV)

11.37a 12.41b 10.60c

Fig. 1, we observe, as expected, a significant charge depletion in the interatomic region accompanied by a density increase along the z axis outside the bond region. The ESCD curve provides a quantitative picture of the amount and spatial extent of the electron displacement. Notice how the curve is positive to the left of the bond midpoint and, symmetrically, negative to the right of it, indicating electron displacement towards the left and right, respectively. The electron depletion (negative slope of the curve) extends beyond the nuclear position, as the curve reaches a maximum in magnitude of ∼0.35 e− at about 0.5 Å away from each nucleus, meaning that about 0.7 electrons have moved out of the region enclosed between these two extrema. Similarly, and perhaps less widely acknowledged, the ESCD curve in the top right panel of Fig. 1 reveals how charge is displaced radially away from the internuclear axis and provides a precise measure of the amount and extent of this flow. In particular, the charge tends to decrease in a region around the molecular axis characterized by a radius of about 0.6 Å, to be then accumulated in a shell extending up to 3.5 Å from the bond, with a net charge decrease of about 0.32 e− . (A partial cylindrical integration of the density difference, which we shall present elsewhere, would provide a more appropriate picture of the radial displacement from the bond axis.) In this first example, to complement the ESCD curves, we also show in the lower panels of Fig. 1 the curve derivatives, i.e., the density difference integrated along the directions perpendicular to the reference axis. These help to gain a more quantitative view of the information provided by the contour plots shown behind the ESCD curves. For the interested reader, analogous ESCD derivatives for all the remaining systems discussed in this paper are reported in the supplementary material.107 Moving to a discussion of the lowest triplet 3 B state which, in a pure MO description, differs from the singlet state only for the spin component, we see that the density difference plots in Fig. 2 describe a qualitatively entirely similar pattern to those discussed above. The ESCD curves, however, evidence clearly a quantitatively much less pronounced electron displacement, both in quantity and spatial extent. The central electron loss along the z axis is about 0.5 e− and it is only about 0.15 e− along an orthogonal axis. These findings may be rationalized by resorting to a valence bond formulation: the triplet state has covalent character while the singlet is described by ionic structures, characterized by more pronounced charge separation.108, 109 We conclude this section by discussing the case of a σ → π excitation, to the 1 C 1 u 2pπ state. This state, in the orbital picture, can be described as the promotion of one electron from the σ g orbital to the lowest π u MO (2pπ u ).110 The discussion reported here concerns the case in which the x component of the two π u orbitals is occupied in the excited state. The partial Rydberg nature of this excited state has been already analyzed both from the experimental and theoretical point of view.111–113 The addition to the basis of a set of molecule-centered diffuse 8s8p8d functions, providing for a proper description of the low-energy Rydberg states, was found to be crucial to accurately reproduce the experimental results. Looking at the density difference plots of Fig. 3, we can immediately recognize the π nature of the transition,

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0.4

0.4

0.2

0.2

Δq (e)

Δq (e)

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−0.2

−0.4

−0.4

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°) Δρ (e/A

°) Δρ (e/A

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0.0

−0.2

−0.4

−0.4 −2

0

2

4

2

4

0.0

−0.2

−4

0 °) x (A

°) z (A

2

4

°) z (A

−4

−2

0 °) x (A

FIG. 1. Top: Isodensity contour plots of the electron density change and ESCD curves for the X 1 g → 1 B 1 u+ excitation of H2 along the internuclear z axis (left panel) and the x axis (right panel). Blue/red surfaces denote charge accumulation/depletion. The density value at the surfaces is ±0.001 e a.u.−3 . Bottom: Corresponding electron density differences integrated along the directions perpendicular to the given axis.

as the main charge rearrangements take place along an axis (x in this case) perpendicular to the bond axis. The picture shows a significant CD away from the bond axis, which the q(x) function quantifies in about 0.72 e− . This is also characterized by a pronounced spatial extension, remaining appreciable even beyond 4 Å from the internuclear axis, further attesting to the partial Rydberg character of the excited state. Looking at the curve integrated along the z axis, we note again that the CD function makes visually evident and quantifies information not otherwise easily extracted from the density difference, namely an electron flow of about 0.36 e− away from the bond region and towards the external side of the hydrogen atoms along the internuclear axis. This depletion, only about

half that in the perpendicular direction and much less spatially extended, is also to be ascribed to the antibonding and partial Rydberg character of the excited state. Practically equivalent information can be extracted by analyzing the CD curve integrated along the y axis, perpendicular itself to the main charge movement.

B. Formaldehyde: The first 1 A2 excited state

The spectroscopic properties of the formaldehyde molecule have been widely investigated both from the experimental and from the theoretical point of view.51 In particular,

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0.4

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0.2

0.2

Δq (e)

Δq (e)

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0.0

−0.2

−0.2

−0.4

−0.4

−4

−2

0

2

−4

4

−2

0

2

4

°) x (A

°) z (A

FIG. 2. Isodensity contour plots of the electron density change and ESCD curves for the X 1 g → 3 B 3 u+ excitation of H2 along the internuclear z axis (left panel) and the x axis (right panel). Blue/red surfaces denote charge accumulation/depletion. The density value at the surfaces is ±0.001 e a.u.−3 .

(a)

(b)

(c)

FIG. 3. Isodensity contour plots of the electron density change and ESCD curves for the X → 1 C excitation of H2 , respectively, along the internuclear z axis (a), the x (b), and the y (c) perpendicular axes. Blue/red surfaces denote charge accumulation/depletion. The density value at the surfaces is ±0.001 e a.u.−3 .

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TABLE II. Experimental and calculated excitation energies ω for the lowest state of formaldehyde.

0.25

1A 2

Method

ω (eV)

NEVPT2-SC NEVPT2-PC NEVPT3 Experimental

4.07 4.07 4.05 3.79a –4.07b

CASSCF(12,10) 2° order corrected density

0.20

b

Δq (e)

a

0.15

Reference 117. Reference 118.

the lowest singlet excited state (1 A2 in the C2v point group), involving the excitation of one electron from the oxygen lone pair to the π * orbital, 1 (n → π *), has attracted much attention because of its central role in the photochemical decomposition of H2 CO in H2 +CO and H+HCO.114 We performed very accurate second and third order NEVPT calculations to reproduce the excitation energy of the 1 A2 state. Following the indication of previously published works,115, 116 the zeroorder wave functions of both the ground and excited states have been obtained by performing state-specific CASSCF calculations, with an active space comprising the 10 valence orbitals [(3-7)a1 , (1-2)b1 , and (1-3)b2 ] and the 12 valence electrons (the molecule lies in the yz plane, with the C and O atoms on the z axis). For this system, we compared the electron densities obtained at the CASSCF level (zero order) with those corrected at the second order of perturbation. The augcc-pVQZ basis set has been used to investigate the system,83 and all the calculations have been performed with the MOLPRO 2010 software package,85 interfaced with a standalone SC-NEVPT3 code.74 The computed excitation energies are reported in Table II together with the corresponding experimental value, showing very good agreement. A corresponding assessment of the convergence and accuracy of the computed density and density difference may be performed by examining the ground- and excited-state dipole moments and the CD curves obtained at different levels of theory. The dipole moments computed at the CASSCF level and using the second order corrected densities are reported in Table III, together with experimental values, while the corresponding CD curves are shown in Fig. 4. The computed dipole moments show satisfactory convergence and agreement with the experimental data, and—more important for our purposes— sufficient convergence of the density change accompanying the excitation is clearly attested by the negligible difference between the CD curves. Based on these results, for this and for all the other investigated molecules we shall only discuss the CD analysis of the relevant CAS/RAS-SCF electron densities. TABLE III. Experimental and calculated dipole moments for the ground and the 1 A2 excited state of the formaldehyde molecule. Density

μGS (D)

μexc (D)

CASSCF(12,10) 2◦ -order Experimentala

− 2.488 − 2.484 − 2.34

− 1.292 − 1.280 − 1.56

a

Reference 119.

0.10

0.05

0.00

−2

−1

0

1

2

3

°) z (A

FIG. 4. Comparison between the CD curve obtained by CASSCF(12,10) calculations and that corrected at the second order of perturbation for the 1 A2 excitation of formaldehyde.

The density difference plots relative to the first 1 A2 state of formaldehyde and the corresponding CD curves are reported in Fig. 5. As is apparent, the density difference plots characterize well the n → π * nature of the transition: charge depletion is localized on the hydrogens and in the region around the oxygen corresponding to the n orbital, while density accumulation appears in the areas above and below the molecular plane, in proximity of the C and O atoms, corresponding to a π * orbital. The ESCD curves along the three spatial directions provide a detailed description of the charge movements accompanying the electronic excitation. In particular, the curve in Fig. 5(a) shows a net CT (about 0.2 e− ) in the direction from the oxygen atom towards the carbon, demonstrating the expected larger localization of the π * orbital on carbon. Also from the hydrogen side a small CT (0.01 e− ) towards the C atom can be observed. Unlike the Rydberg transition discussed earlier for H2 , the charge displacement tends here to remain confined within the molecular region.116 Turning now to Fig. 5(b), the CD function visualizes the charge movements along the x (π ) direction (perpendicular to the molecular plane). We thus see that about 0.4 e− move away from the molecular plane region (where the n orbital is localized) towards outer areas above and below the plane. Also in this case our approach gives us information about the spatial extent of the electron movements, and thus an estimate of the spatial confinement of the electron-accepting orbital. In particular, we see that the π * orbital of the formaldehyde extends no more than 2 Å above and below the molecular plane. Finally, Fig. 5(c) shows the charge rearrangements taking place in the direction orthogonal to the C–O bond but lying in the molecular plane. We can see here a significant charge displacement towards the carbonyl axis which begins about 1.7 Å to its left and right (approximately in correspondence with the position of the hydrogen atoms) and leads to

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(b)

(a)

(c)

FIG. 5. Isodensity contour plots of the electron density change and ESCD curves for the excitation from the ground to the 1 A2 state of formaldehyde. Blue/red surfaces denote charge accumulation/depletion. The density value at the surfaces is ±0.001 e a.u.−3 . The molecule lies on the yz plane, with the z axis coinciding with the C2 symmetry axis. Panels (a), (b), and (c) represent, respectively, the curves integrated along the z, x, and y axes.

an accumulation of about 0.4 e− in a narrow region around the C–O bond axis.

C. Ethene: The 1 B1u excited state

The excitation to the lowest-energy singlet excited state of ethene (1 B1u in the D2h symmetry group of the molecule, which lies in the yz plane, with the C atoms on the z axis), also termed V state in the Merer-Mulliken notation,120 is qualitatively described as the transfer of one electron from the bonding π orbital (b3u ) to the antibonding π * one (b2g ). Due to its well-known “ionic” nature, which makes it a challenge for most of the high-level quantum chemistry methods, the V state of the C2 H4 molecule has been the subject of a large

series of theoretical studies in the last decades (see Refs. 73 and 121–134 for some recent articles). As mentioned, even the most refined theoretical methods found an unusual difficulty in consistently describing this state, recently Angeli demonstrated135, 136 that the problem is essentially related to the difficulty of including orbital polarization effects in the excited state wave function starting from orbitals optimized for the ground state electron distribution. The proposed strategy to overcome this problem, which provided very accurate excitation energies, relies on the use of large RASSCF calculations to include these orbital polarization effects. Here we adopt the higher level of calculation (RAS 3 with the augQZ basis) proposed in Ref. 135 as our reference, in order to obtain an accurate description of the charge density modifications. By using this zero order wave function we obtained a

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(a)

(b)

(c)

FIG. 6. Isodensity contour plots of the electron density change and ESCD curves for the excitation from the ground to the first 1 B1u state of ethene. Blue/red surfaces denote density increase/decrease. The density value at the surfaces is ±0.001 e a.u.−3 . Panels (a), (b), and (c) show the ESCD curves integrated along the z (C–C bond), x, and y axes, respectively. The molecule lies on the yz plane.

RASSCF/PC-NEVPT2 excitation energy of 7.67 eV in perfect agreement with the accepted experimental band maximum value of 7.66 eV. All the calculations have been performed using the Molcas 6.0 software package.137 The density difference plots and the corresponding ESCD curves are reported in Fig. 6. The density difference plots clearly illustrate the π → π * nature of the transition, characterized by a significant charge depletion in the regions above and below the C–C double bond (where the π orbital is mainly localized) and by electron accumulations along the x direction (perpendicular to the molecular plane) in the areas close to the two carbon atoms. Notably, the density difference plots give evidence of the σ polarization effects discussed by Angeli in Refs. 135 and 136. Indeed, one can see a significant charge depletion area close to the hydrogen atoms accompanied by electron accumulation in the region of the C–C double bond.

Also in this case a more quantitative picture of the nature of the excitation can be attained by inspection of the CD curve integrated along the three spatial directions. Looking at Fig. 6(a), we see that about 0.3 e− move away from the region between the carbon atoms and accumulate to the left and right sides of the C–C bond up to about 3.0 Å from the C–C midpoint. Note that the slight slope changes evident at about 1.8 Å are a signature of the H σ polarization mentioned above. Interesting information can be also gained from the CD curve integrated along the x axis perpendicular to the molecular plane. Because of the larger extension of the π * orbital in this direction compared to the π one, about 0.28 e− are globally displaced from the region closer to the plane to a region extending from about 1 to 4 Å above and below it. On a finer level of detail, the ESCD plot also shows a small local charge accumulation near the molecular plane (about 20 me− ), which is again clear evidence of the σ polarization

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of the C–C double bond. Finally, the CD curve along the y axis (lying on the molecular plane orthogonal to the C–C axis) reveals only a small charge polarization, amounting to about 20 me− in proximity of the hydrogen positions.

D. Pyrrole: The 3px Rydberg state

Here we test the reliability of our ESCD approach in the characterization of Rydberg-type excited states, selecting the pyrrole molecule as a prototype example of a small aromatic molecule showing intense Rydberg series overlaid to the valence bands. The study of the spectroscopic properties of this molecule has attracted a great interest because of the difficulties found in the unambiguous assignment of its spectral features as well as for the role that this system plays as a structural unit of many biological and pharmaceutical compounds. For these reasons the electronic absorption spectrum of pyrrole, in the region between 5 and 8 eV, has been widely investigated by both theoretical138–149 and experimental studies.138, 150–159 Here we shall consider only the 3px 1 A1 Rydberg state of pyrrole. The system has a C2v symmetry and it is placed in the yz plane with the z axis as the C2 symmetry axis; the excited state considered involves the promotion of one electron from the π 2b1 orbital to the 3px (a2 symmetry) Rydberg orbital. In this case all the calculations were performed with a contracted ANO-L basis set160 with the contraction scheme C,N[4s3p1d] and H[2s1p]. In order to describe the Rydberg molecular orbitals, the above mentioned valence basis set was augmented with moleculecentered diffuse functions (the same center as in Ref. 144 was used). These basis functions are obtained by contraction of a set of 8s8p8d Gaussian primitives, whose exponents were optimized as described by Kaufmann et al.84 The contraction coefficients were computed following the methodology developed by Roos et al.161 with a contraction scheme [1s1p1d]. To make the comparison between our results and those of the previous ab initio studies more meaningful, the vertical excitation energies were computed using the experimental geometry162 used in Refs. 139 and 144. The electron densities were obtained from average CASSCF calculations using the Molcas 6.0 software package,137 averaging over five states of the same symmetry. The active space used for these calculations (0503) consists of the five (0302) valence π orbitals and three (0201) Rydberg-type orbitals.149 Pastore et al.149 demonstrated the accuracy of this approach in describing this type of excited states, with a SC-NEVPT3 excitation energy of 6.83 eV, in excellent agreement with the results obtained by the most accurate theoretical methods (6.65–6.94 eV).138–149 The density difference plots and the ESCD curves relative to the 2b1 → 3px excitation are reported in Fig. 7. From the density difference plots, we can see a significant charge depletion in the regions above and below the molecular plane at the site of the N atom and of the two external carbons, where the π 2b1 MO is localized, while charge accumulations appear above and below the two central C atoms, outlining the expected picture of the two 3px orbitals.

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The ESCD functions clearly highlight the Rydberg (diffuse) nature of the electron-accepting orbital. In Fig. 7(a), where the ESCD axis runs in the direction from the external C–C bond towards the nitrogen atom, we notice that the curve is negative everywhere, describing a flow of charge (about 0.4 e− ) from the external carbons of the pyrrole ring towards the nitrogen end. As can be seen, CD extends appreciably up to about 3 Å to the right of the nitrogen atom, as a consequence of the very diffuse nature of the 3px orbital. In fact, an appreciable diffuse character can be observed also in the donor 2b1 orbital, as the electron depletion starts to become significant, already 4 Å to the left of the two external carbons. Useful information can be extracted also from the CD curve integrated along the x axis, running perpendicular to the molecular plane. Here the CD provides a measure of the extension of the molecular orbitals involved in the excitation out of the molecular plane. We see that about 0.16 e− are displaced from the region closer to the pyrrole ring towards outer regions above and below the plane, extending as far as about 5 Å from it. This provides an estimate of the extent of the accepting Rydberg orbital along the x axis. Finally, in Fig. 7(c), the CD curve along the axis running across the molecule on the ring plane shows a charge depletion area of about 0.2 e− in the middle of the ring and a corresponding accumulation in the outer regions. The extent of the receiving Rydberg orbital appears clearly more limited in this direction.

E. Spiro cation: A pure CT state

We conclude by discussing the application of the ESCD approach to a pure CT state, which of course represents the most interesting test case for a method looking at the spatial distribution of the electron density changes. We test the method on a small mixed-valence π -σ -π compound, indicated hereafter with “Spiro” (5,5 (4H,4H )spirobi[ciclopenta[c]pyrrole]2,2 ,6,6 tetrahydrocation). Due to its relative small size, which allows fairly accurate calculations, this system has attracted the attention of several theoretical research groups.163–169 In the domain of the intramolecular electron transfer, mixed-valence compounds play a relevant role as simple systems suitable for understanding CT phenomena.170–175 The Spiro molecule consists of two pyrrolic units (π systems), lying on two perpendicular planes, connected by a spirocicloalkane rigid σ bridge. The symmetry of the neutral molecule would be D2d , with the C2 axis (the z axis) passing through the two N atoms, but if an electron is removed from the system, the positive charge tends to localize on the left or on the right pyrrolic unit, distorting the symmetry and giving rise to two equivalent C2v minima. For our investigations we used the geometries optimized for the cation in Refs. 167 and 168 at the restricted open shell Hartree-Fock level with a triple zeta plus polarization (TZP) atomic natural orbital (ANO) basis set, fixing the system in one of the two C2v minima positions (corresponding to a ξ value of 0.5 as described in Ref. 167). The two states involved in the CT process belong to the A2 symmetry, corresponding to the ground and to the first excited state of the cation.

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(b)

(a)

(c)

FIG. 7. Isodensity contour plots of the electron density change and ESCD curves for the excitation from the ground to the 3px Rydberg state of pyrrole. Blue/red surfaces denote charge increase/decrease. The density value at the surfaces is ±0.001 e a.u.−3 . Panels (a), (b), and (c) show the CD curves integrated along the z, x, and y axes, respectively. The molecule lies on the yz plane.

The electron densities used to perform the CD analysis have been obtained by state-averaged CASSCF calculations performed for the two 2 A2 states with the MOLPRO 2010 software package,85 by using the aug-cc-pVTZ basis set.83, 176 The active space used for these calculations is that named CAS(7/4) in Ref. 168 and yields a PC-NEVPT2 excitation energy for the investigated transition of 36.05 kJ/mol in good agreement with the results reported by Pastore et al. in Ref. 168 and references therein. The density difference plots and the CD curves for the 12 A2 → 22 A2 excitation of the Spiro cation are reported in Fig. 8. Looking at the density difference plots, one sees immediately a qualitative illustration of the ET process, as an evident charge depletion on one of the two pyrrolic units as-

sociated with a comparable density increase on the other homologous moiety. Looking at the CD curves, we can exactly quantify this ET. In particular, the curve integrated along the principal C2 axis (z, Fig. 8(a)) shows that more or less exactly one electron is lost from the pyrrolic unit on the right and is accumulated on the corresponding moiety on the left. Note how the ESCD function remains almost completely flat in the region of the σ bridge, evidencing that this part of the molecule only functions as the ET mediator, without undergoing itself any charge accumulation or depletion. The CD curves integrated along the two perpendicular directions (x and y) are not related to the electron transfer during the excitation, and only describe a charge density variation from the donor ring perpendicular to the acceptor one.

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(a)

(b) 1.0

0.5

0.5

Δq (e)

Δq (e)

1.0

0.0

0.0

−6

−4

−2

0

2

4

6

−6

−4

−2

°) z (A

0

2

4

6

°) x (A

(c) 1.0

Δq (e)

0.5

0.0

−6

−4

−2

0

2

4

6

°) y (A

FIG. 8. Isodensity contour plots of the electron density change and ESCD curves for the excitation from the ground to the first 2 A2 state of the Spiro cation. Blue/red surfaces denote charge accumulation/depletion. The density value at the surfaces is ±0.001 e a.u.−3 . Panels (a), (b), and (c) show the CD curves integrated along the z, x, and y axes, respectively.

V. CONCLUSIONS

In this paper we extended the charge displacement analysis to the investigation of the charge shifts occurring during an electronic excitation. This has been possible by simply introducing in the original formalism the electron density difference between the excited and the ground state of the investigated system. The new approach allows an accurate and insightful characterization of electronic transitions, delivering reliable quantitative information about the charge rearrangements taking place during the excitation across the entire molecular space, thus extending the qualitative information provided by other commonly used methods. Such ESCD analysis, applicable to electron densities evaluated at any level of calculation, provides estimates of both the amount of charge that moves during the electronic excitation and the spatial extension of these charge movements, also capturing the small local details of the charge rearrangements due to the excitation process.

We have applied the new approach to a series of systems using very accurate electron densities evaluated by benchmark calculations. The results demonstrated that the analysis of the CD curves in the three spatial directions permits to unambiguously characterize valence, ionic, Rydberg, and CT excited states. We would finally like to stress that the proposed method can also usefully be applied as a very sensitive qualitative/quantitative tool to compare the accuracy of different theoretical methods in the description of the electron density changes between the excited and the ground state, providing new helpful insights for their evaluation.

ACKNOWLEDGMENTS

We thank MIUR-PRIN-2010 and FP7-ENERGY-2010 261920 “ESCORT” for financial support.

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