Dipole-Induced Transition Orbitals: A Novel Tool for Investigating

Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 6624−6630

pubs.acs.org/JPCL

Dipole-Induced Transition Orbitals: A Novel Tool for Investigating Optical Transitions in Extended Systems Gunter Hermann,*,† Lukas Eugen Marsoner Steinkasserer,† Beate Paulus,† and Jean Christophe Tremblay*,†,‡ †

Institut für Chemie und Biochemie, Freie Universität Berlin, Takustraße 3, D-14195 Berlin, Germany Laboratoire de Physique et Chimie Théoriques, CNRS-Université de Lorraine, UMR 7019, ICPM, 1 Bd Arago, 57070 Metz, France

‡

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S Supporting Information *

ABSTRACT: Optical absorption spectra for nanostructures and solids can be obtained from the macroscopic dielectric function within the random phase approximation. While experimental spectra can be reproduced with good accuracy, important properties, such as the charge-transfer character associated with a particular transition, are not retrievable. This contribution presents a computationally inexpensive method for the analysis of optical and excitonic properties for extended systems based on solely their electronic ground-state structure. We formulate a perturbative orbital transformation theory based on dipoleinduced transition moments between orbitals, which yields correlated pairs of particle and hole functions. To demonstrate the potency of this new transformation formalism, we investigate the nature of excitations in inorganic molecular complexes and in extended systems. With our method, it is possible to extract mechanistic insights from the transitions observed in the optical spectrum, without requiring explicit calculation of the many-electron excited states.

F

density functional theory (LR-TDDFT). In this approach, the one-particle transition density matrix can be obtained from the knowledge of the excited-state wave functions. Because all available methods require explicit computation of the electronic structure of the desired excited states, they can be calculated at an affordable numerical cost only for small-tomoderate sized systems. This also applies to visualization techniques based on the difference between excited- and ground-state densities. In contrast, our method obviates the need to compute excited states. We use first-order time-dependent perturbation theory to approximate the one-particle transition density matrix from the transition dipole matrix. The latter is computed solely from the orbitals of the electronic ground state of the system. Hence, only a ground-state calculation at the density functional theory (DFT) level is required to shed light on excitonic properties of a molecule. This makes our technique highly scalable and attractive for extended systems. Because of the conceptual similarity to the NTOs and the usage of the transition dipole matrix as a building block for our computational tool, the new set of correlated particle−hole orbitals is termed dipole-induced transition orbitals (DITOs). Although approximate, this new set of correlated particle−hole orbitals is shown to capture the same physics as the NTOs for optical excitations in isolated molecules. We also show how it is possible to describe the optical spectrum and the excitonic

irst-principles calculations of optical properties have become an invaluable tool for experimental physicists and chemists to corroborate as well as to interpret their results. On the other hand, material design enters an era in which highthroughput calculations are essential to systematically screen for new materials with properties tailored toward specific optical applications.1−3 From this perspective, computationally affordable, high-quality descriptors for optical and excitonic properties of materials are highly desirable. While there exist highly scalable methodologies able to reproduce the experimental spectra for extended materials and solids with good accuracy,4 they often do not provide any information about the excitonic nature of a particular transition and the spatial distribution of the corresponding particles and holes. With this work, we address this issue by providing a practical and straightforward analysis technique revealing the physics of optical excitations in molecular and extended systems. To this end, our methodology uses the transition dipole matrix as a measure allowing us to characterize the nature of an optical transition band in terms of correlated pairs of particle and hole functions. Thus, it provides a direct visualization technique for interpreting electronic excitations associated with specific spectral features. General methods based on the unitary transformation of the one-electron density matrix have been put forward in recent years.5−9 An interesting alternative based on the diagonalization of a difference density matrix was also proposed and proved promising for exciton analysis of timedependent processes.10,11 The formalism we are presenting is conceptually similar to the natural transition orbitals (NTOs),12 as applied to linear response time-dependent © 2018 American Chemical Society

Received: July 19, 2018 Accepted: October 30, 2018 Published: October 30, 2018 6624

DOI: 10.1021/acs.jpclett.8b02253 J. Phys. Chem. Lett. 2018, 9, 6624−6630

Letter

The Journal of Physical Chemistry Letters

this unitary orbital transformation is the one-electron transition density matrix between a single determinantal reference state, Ψ(0), and an excited state, Ψ(λ), with matrix elements given by

properties of an extended system with periodic boundary conditions by performing only a ground-state DFT calculation and explicitly avoiding the expensive computation of a multitude of many-electron excited states. Dipole-Induced Transition Orbitals. First-principles calculations of optical absorption spectra for solids are often based on determining the imaginary part of the macroscopic dielectric function. In general, the dielectric function couples an external perturbative electromagnetic field of frequency ω with the electric field in a material, whose response is described by the frequency-dependent dielectric response function. In the random phase approximation (RPA), it is given by 1 χGG (q, ω) = ′ Ω

∑∑∑ k

pq

ss′

× ψqs ′ k|e−ι(q + G)·r|ψps k + q

T(pqλ) = ⟨Ψ(λ)|ap†̂ aq̂ |Ψ(0)⟩

Here, â†p is the creation operator of orbital ψp, and âq denotes the annihilation operator of orbital ψq. A singular value decomposition of T(λ) is then used to yield the hole and particle functions. In general, the transition density matrix T(λ) can be obtained from single-reference electronic structure methods, such as CIS, TDA, or LR-TDDFT. In this case, the indices p and q in eq 4 respectively run over the Nvirt virtual and Nocc occupied orbitals of the ground-state reference, and the transition density matrix takes the form

fqs ′ k − f ps k + q ω + ϵqs ′ k − ϵps k + q + ιη ψqs ′ k|ei(q + G ′)·r′|ψps k + q

ij T (λ) T1,(λN)occ + 2 jj 1, Nocc + 1 jj jj (λ) jj T T2,(λN)occ + 2 j 2, Nocc + 1 (λ) T = jjj jj jj ∂ jj jj (λ) jjT T (λ) k Nocc , Nocc + 1 Nocc , Nocc + 2

(1)

where η is the positive infinitesimal electronic broadening and Ω is the volume of the unit cell. ψqs′k is the single-particle wave function in the spin channel s′ with eigenenergy ϵqs′k and Fermi−Dirac occupation fqs′k. The reciprocal lattice vectors are denoted by G/G′, and k represents a wave vector in the Brillouin zone. Based on eq 1, a simplified expression for the imaginary part of the dielectric function in the optical limit, ∥q∥ = 0+, can be formulated by neglecting local field effects parallel to q and suppressing the k-point dependence 0α[ε(ω)] =

4πη Ω ×

∑∑ pq

ss′

(ω + ϵqs′ − ϵps)2 + η2

ϵps − ϵqs′

2

(2)

Here, êα is the unit-vector along the three spatial directions, α ∈ {x, y, z}. As was shown in ref 4, important numerical savings can be achieved by computing ψqs′ in spin channel s′ as a molecular or crystal orbital, formed as a linear combination of atomic orbitals (LCAO). Accordingly, we refer to this methodology as LCAO-RPA. The method facilitates the computation of optical spectra for extended systems. The LCAO molecular orbitals are directly obtained from a single electronic ground-state calculation, which obviates the need to compute many-body excited states explicitly. The transition dipole moment matrices in eq 2 Dpq , α =

T(pqλ), α ≃

T1,(λN)occ + Nvirt

2π ℏ

2

ψp Eαμα ψq

(5)

δ(ϵp − ϵq − ℏω(λ))

= Nα |Dpq , α |2 δ(ϵp − ϵq − ℏω(λ))

(6)

for an external electric field of frequency ω(λ) and amplitude Eα. The delta distribution ensures energy conservation. Because the field strength is arbitrary, we use Nα as a normalization constant to ensure particle/hole conservation. In contrast to the NTO formalism, the label λ denotes here a band in the LCAO-RPA spectrum with frequency ω(λ). These bands can be uniquely identified by fitting the spectrum obtained from eq 2 to a sum of Lorentzian functions. These Lorentzians define the frequency ω(λ) and the width η(λ) of a specific transition. The delta distribution in eq 6 is thus replaced by a broadened Lorentzian function of width η(λ), i.e., δ(ϵp − ϵq − ℏω(λ)) = [(ϵp − ϵq − ℏω(λ))2 + (η(λ)/2)2]−1. We have found that taking the same value broadening in the delta distribution and to compute the LCAO-RPA spectrum using eq 2, η(λ) = η, does not change the appearance of the particle/ hole functions, as defined below. The singular value decomposition of the resulting perturbative transition density matrix yields two new sets of transformation matrices for each band, λ, as

eα̂ ⟨ψp|∇α |ψq⟩ ϵp − ϵq

yz zz zz zz (λ) μ T2, Nocc + Nvirt zzzz zz zz zz ∏ zz zz μ TN(λocc) , Nocc + Nvirt zz { μ

Note that the transition density matrix is not rigorously defined in LR-TDDFT, but it can be approximated by renormalizing the excitation amplitudes of the associated excited state. Because these single-reference electronic structure methods are usually not affordable for extended systems, we want to specifically avoid the computation of the many-electron excited states and directly describe the excitonic character of an optical transition in the LCAO-RPA spectrum. Recall that the LCAORPA method requires only an electronic ground-state calculation. Because the bands observed in the LCAO-RPA spectrum are dipole-induced transitions, we approximate the elements of the transition density matrix along direction α, T(λ) pq,α, using first-order time-dependent perturbation theory for a dipole-induced transition between two orbitals, ψp and ψq

fqs′ − f ps

eα̂ ⟨ψps|∇α |ψqs′⟩

(4)

(3)

contain the information about the frequency-dependent intensity of the imaginary part of the dielectric function, which in turn is directly related to the optical absorption spectrum. Note that, from here, we have changed to spin-free notation to simplify the appearance of the equations. The physics for the creation of a particle−hole pair via an external field is fully included in the Dα matrices. To describe the excitonic nature of a specific optical transition obtained at the LCAO-RPA level, we propose to formulate a perturbative theory closely related to the concept of natural transition orbitals (NTOs).12 These provide a compact representation of multideterminantal wave functions Ψ(λ) in terms of correlated pairs of particle and hole functions. The central quantity for 6625


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Figure 1. (a) Comparison of particle (blue) and hole (gray) densities between NTOs and (rotationally averaged) DITOs for the optical transition in [Ru(bpy)3]2+ with the largest peak intensity (4.1 eV for LCAO-RPA, and 4.7 eV for LR-TDFFT). The DITOs and NTOs are weighted by a smoothed Lorentzian function around the respective band maximum. The isosurface value for the densities is set to 0.001a−3 0 . (b) Optical absorption spectra calculated by means of LCAO-RPA (blue) and LR-TDDFT (green) and broadened using Lorentzian functions with a width of η = 0.1 eV. Vertical gray lines in both spectra signify the oscillator strengths of the specific transitions for the LR-TDDFT spectra and the squared transition dipole moments of the orbital transitions for the LCAO-RPA spectra. Black crosses mark the maxima of the respective optical band for which the NTOs and DITOs are calculated. The Lorentzian functions used to weight the NTOs and DITOs are depicted as light blue and green curves. For the LCAO-RPA spectra, the highest occupied molecular orbital−lowest unoccupied molecular orbital (HOMO−LUMO) gap is shifted to the excitation energy of the first excited state with a dominant HOMO−LUMO contribution obtained from the LR-TDDFT calculation. (c) Occupied and virtual orbitals with dominant contributions to the selected optical band in both absorption spectra, LCAO-RPA and LR-TDDFT. The isosurface value for the orbitals is set to ±0.01a0−3.

T(αλ) = U(αλ)Σ(αλ)(V (αλ))* γa , α(rH) =

∑

2. Compute the singular value decomposition (SVD) of the transition density matrix to generate the two new unitary transformation matrices. 3. Calculate the particle and hole functions by weighting the occupied and virtual orbitals (from DFT calculation) with the eigenvectors obtained from the SVD, eqs 8 and 9. 4. Calculate of the particle/hole densities using eq 10 by weighting the squared sums of the DITOs with the squared singular values from eq 7.

(7)

U(paλ), αψp(rH)

(8)

(V (aqλ), α)*ψq(rP)

(9)

p ∈ occ

ϕa , α(rP) =

∑ q ∈ virt

(λ) (λ) where ∑(λ) α = diag (σ1,α , σ2,α , ···). We term the new pairs of particle (ϕa,α(rP)) and hole (γa,α(rH)) orbitals as dipoleinduced transition orbitals (DITOs) to reflect their physical origin and highlight the similarities with the related NTO formalism. For representation purposes, it is preferable to plot the electron and hole densities rather than the NTOs or the DITOs directly. These can be computed straightforwardly by squaring the transformed orbitals, eqs 8 and 9, and the singular values obtained from eq 7, respectively, as

For extended systems, independent sets of DITOs along the three spatial directions α = x, y, z can be distinguished experimentally by the polarization of the exciting radiation. For molecular spectra in the gas phase, where orientation plays no role, the density matrix elements in eq 6 are spherically averaged by summing over all three Cartesian contributions. By evaluating the transition dipole matrix Dα in the basis of the orbitals, both the LCAO-RPA spectrum and the character of the bands can be computed efficiently and consistently from a single electronic ground-state calculation. A detailed comparison of the DITO and NTO formalisms is shown in the Supporting Information. Molecular Systems. In light of the previously discussed analogy between DITOs and NTOs, we first compare both methods for a molecular benchmark system. For this purpose, we choose the tris(2,2′-bipyridine) ruthenium(II) complex ([Ru(bpy)3]2+) and analyze its particle and hole densities for the optical transition with the largest peak intensity (4.1 eV for

ρ(λ)(rH) = ∑a (σa(,λα))2 |γa , α(rH)|2 ; ρ(λ)(rP) = ∑a (σa(,λα))2 |ϕa , α(rP)|2 (10)

The calculation of the particle/hole densities based on the DITOs can be summarized as follows. For a given band λ frequency ω(λ) identified in the LCAO-RPA spectrum: 1. Compute the elements of the transition density matrix using the perturbative expression for the dipole-induced transition (cf. eq 6). 6626


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center. Interestingly, the particle density exhibits a more important nodal structure along the bonds of the ligands than the hole density. The same is observed for the particle density obtained from the DITO analysis (top part of Figure 1a). For the DITO hole density, the character is also found to be the same on the ligands as for the NTO, although it is more confined along the bonds and on the atoms in the latter. For both particle and hole, the densities on the metal center is more corrugated in the DITO picture. The very good agreement of the depicted DITO and NTO densities can be understood by looking at the ortbital transitions involved. In the LCAO-RPA spectrum used for the DITO analysis, the dominant contributions below the Lorentzian peak (blue shaded in Figure 1b) are found to be HOMO−3 → LUMO+1 (24%), HOMO−4 → LUMO+2 (24%), and HOMO−5 → LUMO (11%). These are precisely the same orbital transitions as in the many-electron state used for the NTO analysis, with slightly different statistical weights. Note that the higher weight of the last transition is damped by the Lorentzian function in the perturbative expression eq 6. It can be concluded that the perturbative expression, eq 6, provides a good approximation to the transition density matrix associated with the many-electron excited-state wave functions. Despite some minor discrepancies, the DITO are thus expected to exhibit excitonic properties similar to those of the NTOs. This statement is likely to hold in all molecular systems, where eq 6 is a good approximation (see the Supporting Information for another example). The remaining differences between the NTOs and DITOs stem from the random phase approximation, i.e., the neglect of phase information between the different orbital transitions. This approximation is also at the root of the expression for the LCAO-RPA spectrum (eq 2). Thus, the DITO formalism offers a consistent picture of the absorption spectrum and of the excitonic character of the underlying bands. Extended Systems. To demonstrate the capabilities of the DITOs for extended systems, we investigate the excitonic properties of a phenol-sensitized MoS2 surface by elucidating the nature of specific electronic excitations in its optical spectrum. Over the past few years, MoS2 has attracted much attention as an environmentally stable material possessing an attractive optical gap.20−25 Its advantageous properties qualify it for many applications, including the construction of lightweight, ultrathin solar-cells. In light of the success of dye-sensitized solar cells (DSSCs) employing TiO2 substrates,26−30 it is reasonable to assume that the optical properties of MoS2 might be further enhanced by targeted functionalization.31−35 To ascertain the performance of different potential functionalizing groups, a detailed mechanistic understanding of the optical excitations within such structures, i.e., the excitonic character and the degree of charge separation for certain features in the optical spectrum, is indispensable. To obtain this information, DITOs are ideally suited, allowing a clear and physically meaningful visualization and evaluation of the optical excitations, at the numerical cost of a ground-state electronic structure calculation. In the present study, the phenol molecules are chosen as sensitizers for the MoS2 surface. These can be seen as analogues to the catechol dye, which is a typical chromophore used in dye-sensitized solar cells.28,36−38 Four phenol molecules are adsorbed on a 6 × 6 MoS2 supercell. The structure relaxation is performed using the GPAW program39−44 with the libvdwxc45 implementation of the vdW-

LCAO-RPA and 4.7 eV for LR-TDFFT). As the starting point for the determination of the DITOs and the associated LCAORPA absorption spectrum, the electronic ground-state structure for this metal complex is calculated at the density functional theory level using a Def2-SVP basis set13,14 and the PBE015 functional. For the computation of the NTOs, the excited eigenstates of the complex are additionally required. These are obtained by means of a time-dependent DFT calculation in the linear response regime (LR-TDDFT). All quantum chemistry calculations are performed using the TURBOMOLE program package,16 and both the NTOs and the DITOs are evaluated using the open source postprocessing toolbox ORBKIT.17−19 The LCAO-RPA and LR-TDDFT absorption spectra for the [Ru(bpy)3]2+ complex are in good agreement, as can be seen from Figure 1b. The strongest band, marked with a black cross, is found to be somewhat red-shifted in the LCAO-RPA spectrum at 4.1 eV, compared to 4.7 eV at the LR-TDDFT level. We will focus on this strongest absorption peak for the excitonic analysis, as it is the most important to describe the optical activity of this compound. As described above, the DITOs are obtained from the orbital transition dipole moments obtained from the electronic ground state. The gray bars below the blue curve thus represent the orbital transitions used to compute the LCAORPA spectrum, weighted by the magnitude of their transition dipole moment. In contrast, the NTOs are calculated on the basis of the one-electron transition density matrices between the ground state and the LR-TDDFT excited states belonging to the selected absorption band. Therefore, the gray bars under the green curve represent many-electron excited states, weighted by the magnitude of their oscillator strength. For the comparison, both the DITOs and NTOs are weighted with Lorentzian functions around the respective band maximum, which are illustrated as shaded areas in Figure 1b. The corresponding particle and hole densities obtained from the NTOs and DITOs are shown in Figure 1a. At first glance, all qualitative features of the excitons concur perfectly, and a closer inspection reveals marginal differences in the nodal structures around the metal atom, where the DITO densities are more corrugated. To understand the origin of these minor discrepancies, we turn our attention to the orbitals involved in the selected optical transitions. The most important orbitals are illustrated in Figure 1c, and a detailed analysis of their contributions to the many-electron excited states can be found in the Supporting Information. The dominant state at the LRTDDFT level has ∼23% contribution from each of the HOMO−3 → LUMO+1 and HOMO−4 → LUMO+2 transitions and ∼15% contribution from the HOMO−5 → LUMO transition (see Table S1 of the Supporting Information). In the NTO picture (Figure 1a), we can recognize the ring-shaped hole density delocalized on the bipyridine ligands as coming from all three occupied orbitals {HOMO−3, HOMO−4, and HOMO−5}. The hole density localized on the metal center can be only partially explained by orbital HOMO−5, see Figure 1c, as a myriad of other small components contribute to this band (the contributions above 5% listed in Table S2 of the Supporting Information account for only 76% of the excitonic character). The virtual orbitals {LUMO, LUMO+1, and LUMO+2} further confer a ring shape to the particle density on the ligands, and the latter two orbitals are also involved in the density localized on the metal 6627


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Figure 2. (a) Ball-and-stick model of the optimized molecular structure for the phenol-sensitized MoS2 surface optimized at the vdW-DF-CX level. (b) The associated optical absorption spectrum (blue) calculated by means of LCAO-RPA at the LDA-level in comparison to the clean MoS2 surface (light purple). The spectra have been broadened using Lorentzians with a width of η = 0.1 eV. Vertical gray lines correspond to the squared transition dipole moments of the specific orbital transitions. Two optical absorption bands, A and B, are marked for the DITO calculation. The Lorentzian functions for their weighting are illustrated in light blue. (c) Particle (blue) and hole (gray) densities for the two selected optical bands, −7 −3 A and B. The isosurface value of the densities is set to 2 × 10−4 a−3 0 for band A and 10 a0 for band B.

DF-CX functional.46−49 A LCAO representation is employed for the pseudo wave functions with a double-ζ polarized basis set, and a Γ-centered 2 × 2 Monkhorst−Pack grid is used to sample the primitive Brillouin zone. The optimized structure of the prototypical dye-sensitized MoS2 model systems is depicted in Figure 2a. The associated optical absorption spectra are determined at the LCAO-RPA level of theory, as implemented in ORBKIT.17 The required orbitals are extracted from a periodic DFT calculation with TURBOMOLE16 using the LDA functional at the Γ-point. We used the molybdenum basis set proposed by Corà et al.50 together with a Hay−Wadt type ECP,51 while a POB double-ζ polarized basis set52 was used for all other elements. The optical absorption spectra for the clean MoS2 surface and phenolsensitized MoS2 system are shown in Figure 2b. The examination of the excitonic nature of the phenolsensitized MoS2 surface is solely focused on excitations caused by light polarized parallel to the surface, because these proved to be most sensitive toward functionalizations. In contrast, excitations due to light polarized perpendicular to the surface are largely dominated by MoS2-localized transitions. From the optical spectra in Figure 2b, a very interesting feature can be immediately identified stemming from the functionalization of the MoS2 surface. While pristine MoS2 shows the onset of a large optical band at ∼2.5 eV with no discernible feature below that threshold, a second peak appears in the low-energy part of the spectrum because of the functionalization with the phenol molecules. Furthermore, a shoulder below the high-intensity band and a few weaker peaks emerge in the spectrum covering the 1.5−2.5 eV energy region. This in itself represents a highly desirable feature, if the aim is to increase the photovoltaic efficiency of a MoS2-based solar cell by extending the photovoltaic response range. This information is also crucial for optimizing the performance of such a system by, for example, chemically altering the phenol group. However, the spectrum itself provides no information about the spatial distribution of the electronic excitations. To gain more insights into the excitonic nature of the optical bands, we now take advantage of the capabilities of the DITOs for the analysis of the LCAO-RPA spectrum. The associated particle and hole densities are computed for two optical bands

and depicted in Figure 2c as blue and gray isosurfaces. The bands are identified by fitting the spectrum to a sum of Lorentzians. These two bands are chosen to illustrate the marked change in character from the infrared (band labeled “A” at ca. 1.2 eV) to the visible regime (shoulder labeled “B” at ca. 2.8 eV). Each band is composed of a large number of orbital transitions, as depicted by the gray lines below the spectrum in Figure 2b. By examining the particle (blue) and hole (gray) densities shown in Figure 2c, it can be observed that there is indeed a pronounced difference between these two excitations. While the particle density lying within the MoS2 surface is localized close to the phenol groups in the energy region A, it becomes almost evenly distributed over the MoS2 layer in the energy region B. At the same time, the hole density remains highly localized on the phenol groups in both cases. From the visual inspection, band A has a stronger degree of charge separation, and its particle density is more localized in comparison to band B. Thus, band A is expected to be more efficient for producing free charge carriers in the MoS2 substrate. Interestingly, both bands A and B are composed of a myriad of orbital transitions. The main contributions for the hole and particle densities in band A stem mostly from the frontier orbitals: the HOMO−1 → LUMO+2, HOMO−2 → LUMO +3, and HOMO → LUMO transitions yield 35%, 34%, and 23% of the exciton character, respectively (see Table S5 of the Supporting Information for other small contributions). All these orbitals are well localized around the chromophore, which explains the localized character of the hole and particle densities in band A. In band B, we find dominant contributions for the hole density from the HOMO−55 (13%), the HOMO−2 (7% in total), HOMO−1 (7% in total), and HOMO (4% in total), followed by many smaller orbitals contributions at about 1%. As for band A, the dominant (frontier) orbitals explain the localized character of the hole density in band B. In contrast, the particle density in band B is composed of a myriad of virtual orbitals, from LUMO+2 to LUMO+78 (18 important orbitals are reported in Table S6 of the Supporting Information). This large mixture of orbitals renders its analysis tedious, while the particle/hole densities 6628



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depicted in Figure 2c allow at first glance to recognize the nature of the associated particle density. In this work, we present a low-cost approach for analyzing optical and excitonic properties of molecular and extended systems requiring solely knowledge of their electronic ground state. This is achieved in two steps: (i) compute an absorption spectrum at the LCAO-RPA level using the molecular or crystal orbitals from an electronic ground-state calculation, and (ii) use the transition dipole matrix in the basis of these orbitals to approximate the transition density matrix of the spectral bands using first-order time-dependent perturbation theory. We present a few examples demonstrating how correlated pairs of particle and hole functions can be used to complement the spectral data by providing a better physical understanding of optical transitions. Our approach is conceptually similar to other quasi-particle formalisms, but it avoids the computation of the excited-state wave functions altogether. The numerically efficient combination of LCAORPA spectrum and DITO analysis should be particularly valuable as an analytical tool for rapidly screening potentially interesting optical transitions in extended systems. In addition, the DITO quasi-particles could be used for the subsequent solution of the Bethe−Salpether equation to improve the quality of the optical spectrum in promising extended systems (see, for example, ref 53).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.8b02253. Theoretical derivation of the analogy between DITOs and NTOs, detailed analysis of molecular orbital contributions to the DITOs for the molecular system and the extended system, additional comparison between NTOs and DITOs for a further molecular system (PDF)

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Letter

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Gunter Hermann: 0000-0002-0705-2028 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS L.E.M.S. acknowledges financial support from the Deutsche Forschungsgemeinschaft within the Priority Program (SPP) 1459 and the support from the Studienstiftung des deutschen Volkes e.V. and from the International Max Planck Research School “Functional Interfaces in Physics and Chemistry”. J.C.T. and G.H. are grateful to the Deutsche Forschungsgemeinschaft for funding through projects TR1109/2-1 and Pe2297/1-1. The computer facilities of the Freie Universität Berlin (ZEDAT) are acknowledged for computer time. Furthermore, we thank Hans-Christian Hege for providing the ZIBAmira visualization program.54 6629


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