Dipole-Induced Transition Orbitals: A Novel Tool for Investigating

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Spectroscopy and Photochemistry; General Theory

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 J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02253 • Publication Date (Web): 30 Oct 2018 Downloaded from http://pubs.acs.org on October 31, 2018

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The Journal of Physical Chemistry Letters

Dipole-Induced Transition Orbitals A Novel Tool for Investigating Optical Transitions in Extended Systems ∗,†

Gunter Hermann,

Lukas Eugen Marsoner Steinkasserer,

Jean Christophe Tremblay

†Institut

†

Beate Paulus,

†

and

∗,†,‡

für Chemie und Biochemie, Freie Universität Berlin, Takustraÿe 3, D-14195 Berlin, Germany

‡Laboratoire

de Physique et Chimie Théoriques, CNRS-Université de Lorraine, UMR 7019, ICPM, 1 Bd Arago, 57070 METZ, FRANCE

E-mail: [email protected]; [email protected]

Abstract Optical absorption spectra for nano-structures and solids can be obtained from the macroscopic dielectric function within the Random Phase Approximation (RPA). 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 solely based on their electronic ground state structure.

We formulate a perturbative orbital transformation theory

based on dipole-induced transition moments between orbitals, which yields correlated pairs of particle and hole functions. In order 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 1

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mechanistic insights from the transitions observed in the optical spectrum, without requiring explicit calculation of the many-electron excited states.

Graphical TOC Entry

Introduction

First principle 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 high-throughput calculations are essential to systematically screen for new materials with properties tailored towards specic optical applications.

13

In this perspective, computationally aordable, 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 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 specic spectral features. General methods based on

2


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the unitary transformation of the one-electron density matrix have been put forward in recent years.

59

An interesting alternative based on the diagonalization of a dierence density

matrix was also proposed, and proved promising for exciton analysis of time-dependent 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 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. Since all available methods require explicit computation of the electronic structure of the desired excited states, they can only be calculated at an aordable numerical cost for small to moderate size systems. This also applies to visualization techniques based on the dierence between excited and ground state densities. In contrast, our method obviates the need to compute excited states.

We use rst-

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.

Due to 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 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.

3



Dipole-Induced Transition Orbitals

Page 4 of 23

First-principle 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 eld of frequency

ω

with the electric eld in a material, whose response is

described by the frequency-dependent dielectric response function.

In the random phase

approximation, it is given by

fqs0 k − fpsk+q 1 XXX Ω k pq ss0 ω + qs0 k − psk+q + ıη E −ı(q+G)·r

D ı(q+G0 )·r0 × ψqs0 k e ψpsk+q ψqs0 k e ψpsk+q ,

χGG0 (q, ω) =

where cell.

η

is the positive innitesimal electronic broadening, and

ψqs0 k

Ω

is the single-particle wave function in the spin channel

and Fermi-Dirac occupation

fqs0 k .

is the volume of the unit

s0

with eigenenergy

The reciprocal lattice vectors are denoted by

k represents a wave vector in the Brillouin zone.

q

eˆα

and suppressing the

k -point

is the unit-vector along the three spatial directions,

dependence

α ∈ {x, y, z}.

in Ref. 4, important numerical savings can be achieved by computing nel

s0

and

kqk = 0+ , can be formulated

eˆα hψps |∇α | ψqs0 i 2 4πη X X fqs0 − fps . Iα [ε(ω)] = Ω pq ss0 (ω + qs0 − ps )2 + η 2 ps − qs0 Here,

G/G0 ,

qs0 k

Based on Eq. (1), a simplied expression for

the imaginary part of the dielectric function in the optical limit, by neglecting local eld eects parallel to

(1)

ψqs0

(2)

As was shown in spin chan-

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.

4


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The transition dipole moment matrices in Eq. (2)

Dpq,α =

eˆα hψp |∇α | ψq i p − q

(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 eld is fully included in the

Dα

matrices. In order to describe the excitonic nature of a specic 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). pact representation of multi-determinantal wave functions

Ψ(λ)

12

These provide a com-

in terms of correlated pairs

of particle and hole functions. The central quantity for 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

(λ) † (0) a T(λ) ˆp a ˆq Ψ . pq = Ψ

Here,

a ˆ†p

orbital

ψp ,

is the creation operator of orbital

ψq .

A singular value decomposition of

and

T(λ)

a ˆq

(4)

denotes the annihilation operator of

is then used to yield the hole and particle

functions. In general, the transition density matrix

T(λ)

can be obtained from single-reference elec-

tronic structure methods, such as CIS, TDA or LR-TDDFT. In this case, the indices

q

in Eq. (4) respectively run over the

Nvirt

virtual and

5

Nocc


p

and

occupied orbitals of the ground


Page 6 of 23

state reference, and the transition density matrix takes the form



T(λ)

(λ)

(λ)

(λ)

T1,Nocc +2 . . . T1,Nocc +Nvirt  T1,Nocc +1  (λ) (λ)  T (λ) T2,Nocc +2 . . . T2,Nocc +Nvirt 2,Nocc +1  = . ..  . . .   (λ) (λ) (λ) TNocc ,Nocc +1 TNocc ,Nocc +2 . . . TNocc ,Nocc +Nvirt

        

(5)

Note that the transition density matrix is not rigorously dened in LR-TDDFT, but it can be approximated by renormalizing the excitation amplitudes of the associated excited state. Since these single-reference electronic structure methods are usually not aordable for extended systems, we want to specically avoid the computation of the many-electron excited states and directly describe the excitonic character of an optical transition in the LCAORPA spectrum.

Recall that the LCAO-RPA method requires only an electronic ground

state calculation. Since the bands observed in the LCAO-RPA spectrum are dipole-induced transitions, we approximate the elements of the transition density matrix along direction

(λ)

Tpq,α ,

α,

using rst-order time-dependent perturbation theory for a dipole-induced transition

between two orbitals,

ψp

and

ψq , 2π |hψp |Eα µα |ψq i|2 δ p − q − ~ω (λ) ~ =Nα |Dpq,α |2 δ p − q − ~ω (λ) ,

T(λ) pq,α '

for an external electric eld of frequency

ω (λ)

and amplitude

Eα .

(6)

The delta distribution en-

sures energy conservation. Since the eld 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 identied by tting the spectrum obtained from Eq. (2) to a sum of Lorentzian functions. These Lorentzians dene the frequency

ω (λ)

and the width

η (λ)

of a specic tran-

sition. The delta distribution in Eq. (6) is thus replaced by a broadened Lorentzian function

6


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of width

η (λ) ,

i.e.,

−1 δ p − q − ~ω (λ) = (p − q − ~ω (λ) )2 + (η (λ) /2)2 .

We have found

that taking the same value broadening in the delta distribution and to compute the LCAORPA spectrum using Eq. (2),

η (λ) = η ,

does not change the appearance of the particle/hole

functions, as dened below. The singular value decomposition of the resulting perturbative transition density matrix yields two new sets of transformation matrices for each band,

λ,

as

∗ (λ) T(λ) = U(λ) Vα(λ) α α Σα X γa,α (rH ) = U(λ) pa,α ψp (rH )

(7) (8)

p∈occ

φa,α (rP ) =

X

(λ) Vaq,α

∗

ψq (rP ),

(9)

q∈virt

where

(λ)

Σα =

(λ) (λ) diag(σ1,α , σ2,α , . . . ).

We term the new pairs of particle (γa,α (rH )) and hole

(φa,α (rP )) orbitals as dipole-induced transition orbitals (DITOs) to reect 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

ρ(λ) (rh ) =

X

(λ) σa,α

2

|γa,α (rh )|2

ρ(λ) (rP ) =

;

X

a

(λ) σa,α

2

|φa,α (rP )|2

(10)

a

The calculation of the particle/densities based on the DITOs can be summarized as follows. For a given band

λ

frequency

ω (λ)

identied 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)). 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

7



Page 8 of 23

(from DFT calculation) with the eigenvectors obtained from the SVD, Eqs. (8,9). 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).

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 radia-

tion. 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 eciently and consistently from a single electronic ground state calculation. A detailed comparison of the DITO and NTO formalisms is shown in the Supporting Information.

Results and Discussion

Molecular Systems.

In light of the previously discussed analogy between DITOs and NTOs, we rst 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 LCAO-RPA, and 4.7 eV for LR-TDFFT). As the starting point for the determination of the DITOs and the associated LCAO-RPA absorption spectrum, the electronic ground state structure for this metal complex is calculated at the density functional theory level (DFT) using a Def2-SVP basis set

13,14

and the PBE0

15

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 calcu-

Turbomole program package, 16 and both the NTOs and 1719 the DITOs are evaluated using the open source post-processing toolbox ORBKIT. lations are performed using the

The LCAO-RPA and LR-TDDFT absorption spectra for the [Ru(bpy)3 ]

2+

complex are

in good agreement, as can be seen from Fig. 1(b). The strongest band, marked with a black

8


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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 LCAO-RPA spectrum, weighted by the magnitude of their transition dipole moment. On the contrary, the NTOs are calculated on the basis of the one-electron transition density matrices between the ground state and the LRTDDFT 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 Fig. 1(b). The corresponding particle and hole densities obtained from the NTOs and DITOs are shown in Fig. 1(a). At rst glance, all qualitative features of the excitons concur perfectly, and a closer inspection reveals marginal dierences 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 Fig. 1(c), 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 LR-TDDFT 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 (Fig. 1(a)), we can recognize the ring-

shaped hole density delocalized on the bipyridine ligands as coming from all three occupied orbitals {HOMO-3, HOMO-4, HOMO-5}.

The hole density localized on the metal center

can only be partially explained by orbital HOMO-5, see Fig. 1(c), as a myriad of other

9



small components contribute to this band (the contributions above 5% listed in Table S2 of the Supporting Information account only for 76% of the excitonic character). The virtual orbitals {LUMO, LUMO+1, 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 center. Interestingly, the particle density exhibit 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 Fig. 1(a)). 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 conned 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 are in Fig. 1(b)) 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 dierent 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 similar excitonic properties as the NTOs. This statement is likely to hold in all molecular systems, where Eq. (6) is a good approximation (see Supporting Information for another example). The remaining dierences between the NTOs and DITOs stem from the Random Phase Approximation, i.e., the neglect of phase information between the dierent orbital transitions. This approximation is also at the root of the expression for the LCAO-RPA spectrum, Eq. (2). Thus, the DITO formalism oers a consistent picture of the absorption spectrum and of the excitonic character of the underlying bands.

10


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Extended Systems.

In order 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 specic 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.

2025

Its advantageous properties qualies it for many applications including the

construction of light-weight, ultra-thin solar-cells. In light of the success of dye-sensitized solar cells (DSSC) employing TiO2 substrates,

2630

it is reasonable to assume that the optical properties of MoS2 might be further enhanced by targeted functionalization.

3135

In order to ascertain the performance of dierent 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 these 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. supercell. libvdwxc

45

28,3638

Four phenol molecules are adsorbed on a

The structure relaxation is performed using the implementation of the vdW-DF-CX functional.

6×6

GPAW program 3944 with the

4649

A LCAO representation is

employed for the pseudo wave functions with a double-ζ polarized basis set and a

2×2

MoS2

Γ-centered

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 Fig. 2(a). The associated optical absorption spectra are determined at the LCAO-RPA level of theory,

ORBKIT. 17 The required orbitals are extracted from a periodic DFT 16 calculation with Turbomole using the LDA functional at the Γ-point. We used the

as implemented in

molybdenum basis set proposed by Furio Corà

11

50

together with a Hay-Wadt type ECP,


51


while a POB double-ζ polarized basis set

52

Page 12 of 23

was used for all other elements.

The optical

absorption spectra for the clean MoS2 surface and phenol-sensitized MoS2 system are shown in Fig 2(b). The examination of the excitonic nature of the phenol-sensitized MoS2 surface, is solely focused on excitations caused by light polarized parallel to the surface, since these proved to be most sensitive towards functionalizations. In contrast, excitations due to light polarized perpendicular to the surface are largely dominated by MoS2 -localized transitions.

From

the optical spectra in Fig. 2(b), a very interesting feature can be immediately identied 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 due to 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 eciency 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, e.g., chemically altering the phenol group. However, the spectrum itself provides no information about the spatial distribution of the electronic excitations. In order 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 Fig. 2(c) as blue and gray isosurfaces. The bands are identied by tting 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 circa 1.2 eV) to the visible regime (shoulder labeled B at circa 2.8 eV). Each band is composed of a large number of orbital transitions, as depicted by the gray lines below the spectrum in Fig. 2(b). By examining the particle (blue) and hole (gray) densities shown in Fig. 2(c), it can be observed that there is indeed a

12


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pronounced dierence 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, the band A has a stronger degree of charge separation and its particle and hole density is more localized in comparison to band B. Thus, the band A is expected to be more ecient 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 respectively 35%, 34%, and 23% of the exciton character, respectively (see Tab. 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 particel densities in band A. In band B, we nd dominant contributions for the hole density from the HOMO-55 (13%). the HOMO-2 (7% in total), HOMO-1 (7% in total), and (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. On the contrary, 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 Tab. S6 of the Supporting Information).

This large mixture of orbitals renders its analysis tedious, while the DITO

depicted in Fig. 2(c) allows at rst glance to recognize the nature of the associated particle density.

Conclusions

In this work, we present a low-cost approach for analyzing optical and ex-

citonic properties of molecular and extended systems solely requiring 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

13



Page 14 of 23

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 rst-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 computation of the excited state wave functions altogether. The numerically ecient combination of LCAO-RPA 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 e.g. Ref. 53).

Acknowledgement L.E.M.S. acknowledges nancial 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. Further, we thank Hans-Christian Hege for providing the ZIBAmira visualization program.

Supporting Information Available

54

Theoretical derivation of the analogy between DI-

TOs 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

14


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References (1) Balanay, M. P.; Kim, D. H. DFT/TD-DFT Molecular Design of Porphyrin Analogues for Use in Dye-Sensitized Solar Cells. Phys. Chem. Chem. Phys. 2008, 10, 51215127.

(2) Angelis, F. D.; Fantacci, S.; Selloni, A. Alignment of the Dye's Molecular Levels with the TiO2 Band Edges in Dye-Sensitized Solar Cells: A DFTTDDFT Study. Nanotechnology

2008,

19, 424002.

(3) Singh, A. K.; Mathew, K.; Zhuang, H. L.; Hennig, R. G. Computational Screening of 2D Materials for Photocatalysis. J. Phys. Chem. Lett. 2015, 6, 10871098.

(4) Glanzmann, L. N.; Mowbray, D. J.; del Valle, D. G. F.; Scotognella, F.; Lanzani, G.; Rubio, A. Photoinduced Absorption within Single-Walled Carbon Nanotube Systems. J. Phys. Chem. C

2016,

120, 19261935.

(5) Plasser, F.; Wormit, M.; Dreuw, A. New Tools for the Systematic Analysis and Visualization of Electronic Excitations. I. Formalism. J. Chem. Phys. 2014, 141, 024106.

(6) Plasser, F.; Bäppler, S. A.; Wormit, M.; Dreuw, A. New Tools for the Systematic Analysis and Visualization of Electronic Excitations. II. Applications. J. Chem. Phys.

2014,

141, 024107.

(7) Bäppler, S. A.; Plasser, F.; Wormit, M.; Dreuw, A. Exciton Analysis of Many-Body Wave Functions: Bridging the Gap between the Quasiparticle and Molecular Orbital Pictures. Phys. Rev. A 2014, 90, 052521.

(8) Plasser, F.; Thomitzni, B.; Bäppler, S. A.; Wenzel, J.; Rehn, D. R.; Wormit, M.; Dreuw, A. Statistical Analysis of Electronic Excitation Processes: Spatial Location, Compactness, Charge Transfer, and Electron-Hole Correlation. J. Comput. Chem.

2015,

36, 16091620.

15



(9) Mewes, S. A.; Plasser, F.; Dreuw, A. Communication:

Page 16 of 23

Exciton Analysis in Time-

Dependent Density Functional Theory: How Functionals Shape Excited-State Characters. J. Chem. Phys. 2015, 143, 171101.

(10) Dutoi, A. D.; Wormit, M.; Cederbaum, L. S. Ultrafast Charge Separation Driven by Dierential Particle and Hole Mobilities. J. Chem. Phys. 2011, 134, 024303.

(11) Dutoi, A. D. Visualising Many-Body Electron Dynamics Using One-Body Densities and Orbitals. Mol. Phys. 2013, 112, 111.

(12) Martin, R. L. Natural Transition Orbitals. J. Chem. Phys. 2003, 118, 47754777.

(13) Schäfer, A.; Horn, H.; Ahlrichs, R. Fully Optimized Contracted Gaussian Basis Sets for Atoms Li to Kr. J. Chem. Phys. 1992, 97, 25712577.

(14) Weigend, F.; Ahlrichs, R. Balanced Basis Sets of Split Valence, Triple Zeta Valence and Quadruple Zeta Valence Quality for H to Rn: Design and Assessment of Accuracy. Phys. Chem. Chem. Phys.

2005,

7, 32973305.

(15) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0 Model. J. Chem. Phys. 1999, 110, 61586170.

(16) TURBOMOLE

V7.2

2017,

A

Development

of

University

of

Karlsruhe

and

Forschungszentrum Karlsruhe GmbH, 1989-2007, TURBOMOLE GmbH, since 2007; Available from

http://www.turbomole.com. (17) Hermann, G.; Pohl, V.; Tremblay, J. C.; Paulus, B.; Hege, H.-C.; Schild, A. ORBKIT: A Modular Python Toolbox for Cross-Platform Postprocessing of Quantum Chemical Wavefunction Data. J. Comput. Chem. 2016, 37, 15111520.

16


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


(18) Pohl, V.; Hermann, G.; Tremblay, J. C. An Open-Source Framework for Analyzing

N-

Electron Dynamics. I. Multideterminantal Wave Functions. J. Comput. Chem. 2017, 38, 15151527.

(19) Hermann, G.; Pohl, V.; Tremblay, J. C. An Open-Source Framework for Analyzing

N-

Electron Dynamics. II. Hybrid Density Functional Theory/Conguration Interaction Methodology. J. Comput. Chem. 2017, 38, 23782387.

(20) Ramakrishna Matte, H.; Gomathi, A.; Manna, A.; Late, D.; Datta, R.; Pati, S.; Rao, C. MoS2 and WS2 Analogues of Graphene. Angew. Chem. Int. Ed. 2010, 49, 40594062.

(21) Radisavljevic, B.; Radenovic, A.; Brivio, J.; Giacometti, V.; Kis, A. Single-layer MoS2 Transistors. Nat. Nanotechnol. 2011, 6, 147150.

(22) Wang, Q. H.; Kalantar-Zadeh, K.; Kis, A.; Coleman, J. N.; Strano, M. S. Electronics and Optoelectronics of Two-Dimensional Transition Metal Dichalcogenides. Nat. Nanotechnol.

2012,

7, 699712.

(23) Lopez-Sanchez, O.; Lembke, D.; Kayci, M.; Radenovic, A.; Kis, A. Ultrasensitive Photodetectors Based on Monolayer MoS2 . Nat. Nanotechnol. 2013, 8, 497501.

(24) Addou, R.; McDonnell, S.; Barrera, D.; Guo, Z.; Azcatl, A.; Wang, J.; Zhu, H.; Hinkle, C. L.; Quevedo-Lopez, M.; Alshareef, H. N.; Colombo, L.; Hsu, J. W. P.; Wallace, R. M. Impurities and Electronic Property Variations of Natural MoS2 Crystal Surfaces. ACS Nano 2015, 9, 91249133.

(25) Rasmussen, F. A.; Thygesen, K. S. Computational 2D Materials Database: Electronic Structure of Transition-Metal Dichalcogenides and Oxides. J. Phys. Chem. C 2015, 119, 1316913183.

(26) O'Regan, B.; Grätzel, M. A Low-Cost, High-Eciency Solar Cell Based on DyeSensitized Colloidal TiO2 Films. Nature 1991, 353, 737740.

17



(27) Hagfeldt, A.; Grätzel, M. Light-Induced Redox Reactions in Nanocrystalline Systems. Chem. Rev.

1995,

95, 4968.

(28) Duncan, W. R.; Prezhdo, O. V. Theoretical Studies of Photoinduced Electron Transfer in Dye-Sensitized TiO2 . Annu. Rev. Phys. Chem. 2007, 58, 143184.

(29) Gomez, T.; Hermann, G.; Zarate, X.; Pérez-Torres, J.; Tremblay, J. Imaging the Ultrafast Photoelectron Transfer Process in Alizarin-TiO2 . Molecules 2015, 20, 1383013853.

(30) Hermann, G.; Tremblay, J. C. Ultrafast Photoelectron Migration in Dye-Sensitized Solar Cells: Phys.

2016,

Inuence of the Binding Mode and Many-Body Interactions. J. Chem. 145, 174704.

(31) Yu, S. H.; Lee, Y.; Jang, S. K.; Kang, J.; Jeon, J.; Lee, C.; Lee, J. Y.; Kim, H.; Hwang, E.; Lee, S.; Cho, J. H. Dye-Sensitized MoS2 Photodetector with Enhanced Spectral Photoresponse. ACS Nano 2014, 8, 82858291.

(32) Jia, T.; Li, M. M. J.; Ye, L.; Wiseman, S.; Liu, G.; Qu, J.; Nakagawa, K.; Tsang, S. C. E. The Remarkable Activity and Stability of a Dye-Sensitized Single Molecular Layer MoS2 Ensemble for Photocatalytic Hydrogen Production. ChemComm 2015, 51, 1349613499.

(33) Huang, Y.; Zheng, W.; Qiu, Y.; Hu, P. Eects of Organic Molecules with Dierent Structures and Absorption Bandwidth on Modulating Photoresponse of MoS2 Photodetector. ACS Appl. Mater. Interfaces 2016, 8, 2336223370.

(34) Nguyen, E. P.; Carey, B. J.; Harrison, C. J.; Atkin, P.; Berean, K. J.; Gaspera, E. D.; Ou, J. Z.; Kaner, R. B.; Kalantar-zadeh, K.; Daeneke, T. Excitation Dependent Bidirectional Electron Transfer in Phthalocyanine-Functionalised MoS2 Nanosheets. Nanoscale

2016,

8, 1627616283.

18


Page 18 of 23

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


(35) Le, H. M.; Bui, V. Q.; Tran, P. H.; Pham-Tran, N.-N.; Kawazoe, Y.; Nguyen-Manh, D. The Prospect of Sensitizing Organic Dyes Attached to the MoS2 Surface:

Physical

Insights from Density Functional Theory Investigations. Chem. Phys. Lett. 2017, 667, 290295.

(36) Duncan, W. R.; Prezhdo, O. V. Electronic Structure and Spectra of Catechol and Alizarin in the Gas Phase and Attached to Titanium. J. Phys. Chem. B 2005, 109, 365373.

(37) de Armas, R. S.; San-Miguel, M.; Oviedo, J.; Sanz, J. F. Direct vs. Indirect Mechanisms for Electron Injection in DSSC: Catechol and Alizarin. Comput. Theor. Chem. 2011, 975, 99105.

(38) Ooyama, Y.; Kanda, M.; Uenaka, K.; Ohshita, J. Eect of Substituents in Catechol Dye Sensitizers on Photovoltaic Performance of Type II Dye-Sensitized Solar Cells. ChemPhysChem

2015,

16, 30493057.

(39) Larsen, A. et al. The Atomic Simulation Environment

−− A Python library for Working

with Atoms. J. Phys. Condens. Matter 2017, 29, 273002.

(40) Mortensen, J. J.; Hansen, L. B.; Jacobsen, K. W. Real-Space Grid Implementation of the Projector Augmented Wave Method. Phys. Rev. B 2005, 71, 035109.

(41) Enkovaara, J. et al. Electronic Structure Calculations with GPAW: A Real-Space Implementation of the Projector Augmented-Wave Method. J. Phys. Condens. Matter

2010,

22, 253202.

(42) Larsen, A. H.; Vanin, M.; Mortensen, J. J.; Thygesen, K. S.; Jacobsen, K. W. Localized Atomic Basis Set in the Projector Augmented Wave Method. Phys. Rev. B 2009, 80, 195112.

19



(43) Marques, M. A.; Oliveira, M. J.; Burnus, T. Libxc: A Library of Exchange and Correlation Functionals for Density Functional Theory. Comput. Phys. Commun. 2012, 183, 22722281.

(44) Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 1795317979.

(45) Larsen, A. H.; Kuisma, M.; Löfgren, J.; Pouillon, Y.; Erhart, P.; Hyldgaard, P. libvdwxc: A Library for ExchangeCorrelation Functionals in the vdW-DF Family. Model. Simul. Mater. Sci. Eng.

2017,

25, 065004.

(46) Berland, K.; Arter, C. A.; Cooper, V. R.; Lee, K.; Lundqvist, B. I.; Schröder, E.; Thonhauser, T.; Hyldgaard, P. Van der Waals Density Functionals Built Upon the Electron-Gas Tradition: Phys.

2014,

Facing the Challenge of Competing Interactions. J. Chem.

140, 18A539.

(47) Berland, K.; Hyldgaard, P. Exchange Functional that Tests the Robustness of the Plasmon Description of the van der Waals Density Functional. Phys. Rev. B 2014, 89, 035412.

(48) Román-Pérez, G.; Soler, J. M. Ecient Implementation of a van der Waals Density Functional:

Application to Double-Wall Carbon Nanotubes. Phys. Rev. Lett. 2009,

103, 096102.

(49) Gharaee, L.; Erhart, P.; Hyldgaard, P. Finite-Temperature Properties of Nonmagnetic Transition Metals: Comparison of the Performance of Constraint-Based Semilocal and Nonlocal Functionals. Phys. Rev. B 2017, 95, 085147.

(50) Corà, F.; Patel, A.; Harrison, N. M.; Roetti, C.; Catlow, C. R. A. An Ab Initio HartreeFock Study of

α-MoO3 .

J. Mater. Chem.

1997,

7, 959967.

(51) Hay, P. J.; Wadt, W. R. Ab Initio Eective Core Potentials for Molecular Calculations. Potentials for the Transition Metal Atoms Sc to Hg. J. Chem. Phys. 1985, 82, 270283.

20


Page 20 of 23

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


(52) Peintinger, M. F.; Oliveira, D. V.; Bredow, T. Consistent Gaussian Basis Sets of TripleZeta Valence with Polarization Quality for Solid-State Calculations. J. Comput. Chem.

2012,

34, 451459.

(53) Onida, G.; Reining, L.; Rubio, A. Electronic Excitations: Density-Functional versus Many-Body Green's-Function Approaches. Rev. Mod. Phys. 2002, 74, 601.

(54) Stalling, D.; Westerho, M.; Hege, H.-C. The Visualization Handbook ; Elsevier, 2005; pp 749767.

21



Figure 1: (a) Comparison of particle (blue) and hole (gray) densities between NTOs and 2+ (rotationally averaged) DITOs for the optical transition in [Ru(bpy)3 ] with the largest peak intensity (4.1 eV for LCAO-RPA, and 4.8 eV for LR-TDFFT). The DITOs and NTOs are weighted by a smoothed Lorentzian function around the respective band maximum. The −3 isosurface value for the densities is set to 0.001 a0 . (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 specic 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 HOMO-LUMO gap is shifted to the excitation energy of the rst 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 bands in both absorption spectra, LCAO-RPA and LR-TDDFT. The −3 isosurface value for the orbitals is set to ±0.01 a0 .

22


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Figure 2:

(a) Ball-and-stick models 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 tran-

sition dipole moments of the specic 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 (red) densities for the two selected −4 −3 optical bands, A and B. The isosurface value of the densities is set to 2 · 10 a0 for band −7 −3 A and 10 a0 for band B.

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Dipole-Induced Transition Orbitals: A Novel Tool for Investigating

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