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Plasmonic Resonances in the Al Cluster: Quantification and Origin of Exciton Collectivity David Casanova, Jon M. Matxain, and Jesus M. Ugalde J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b03210 • Publication Date (Web): 20 May 2016 Downloaded from http://pubs.acs.org on May 22, 2016

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The Journal of Physical Chemistry

Plasmonic Resonances in the Al− 13 Cluster: Quantification and Origin of Exciton Collectivity David Casanova,∗,†,‡ Jon M. Matxain,† and Jesus M. Ugalde† Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU) and Donostia International Physics Center (DIPC), P.K. 1072, 20018 Donostia, Euskadi, Spain. E-mail: [email protected]

∗

To whom correspondence should be addressed Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU) and Donostia International Physics Center (DIPC), P.K. 1072, 20018 Donostia, Euskadi, Spain. ‡ IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Euskadi, Spain. †

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Abstract Recently, plasmonic resonances in molecules, clusters and nanostructures have gathered a lot of attention for their potential range of applicability. Unlike other metal nanostructures, a wide variety of aluminum nanostructures show very promising plasmonic properties. Theoretical and computational investigations helped to understand the nature of collective excitations in such finite systems. However, such theoretical investigations are based on qualitative approaches rather than accurate quantitative methods. In the present work, the collectivity of the low-lying states of Al− 13 are investigated within the time-dependent density functional theory and analyzed through different computational tools. A novel tool, which provides a quantitative index of the collective nature of the electronic excitations, has been introduced, the so called Transition Inverse Participation Ratio (TIPR) index. The obtained results suggest the presence of plasmonic-like transitions in the Al− 13 cluster, and that these transitions are linked to the icosahedral symmetry of the cluster, allowing the rationalization by the simple jellium model. We believe that the present work opens the opportunity for the study and applicability of this type of electronic transition in aluminum clusters and in other related nanostructures.

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Introduction Plasmonics refers to the study of the optical properties of materials and their potential applications in biosensing, 1–3 imaging, 4 photovoltaics, 5 in cancer therapeutics, 6 or in the design and fabrication of metamaterials. 7 This is eminently an interdisciplinary field of research at the interface of chemistry, physics and material science. The plasmon resonance is defined classically as a collective electronic excitation characterized as oscillations of the electronic density caused by restoring forces created by the induced electromagnetic field. The presence of plasmonic states is a common feature in some extended systems, and their study belongs to the solid state physics field. Plasmons have been postulated to arise also in finite systems, from nanostructures to molecular compounds. 8 Concretely, this phenomenon has been theoretically 9 and experimentally studied for a large variety of systems, such as in sodium clusters, 10 metallic nanoparticles, 11 in polycyclic aromatic hydrocarbons 12,13 or in graphene. 14,15 These discoveries open new opportunities in the field, which may have strong implications in the development of new and improved applications. In this context, Al nanocompounds have gained much attention in the last years due to their enhanced plasmonic properties. 16–19 Experimental and theoretical works have shown that Al nanocompounds such as nanoparticles, nanorods or nanodisks exhibit plasmon resonances from the visible to the deep UV regions. In these works, nanostructures as small as 20 nm across were investigated, and the importance of the oxide shell was also analyzed. Due to the size of the nanocompounds, theoretical macroscopic models such as the discrete dipole approximation and finite difference time domain methods were used. In order to use ab initio methods, smaller structures need to be considered. In this vein, the study of the electronic structure nature of plasmon resonances in molecules, clusters and nanoparticles has been computationally addressed, in most of the cases, with time-dependent density functional theory (TDDFT). 20,21 Other approaches employed in the computation of collective excitations are the configuration interaction (CI), 22 3



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the equation-of-motion coupled-cluster, 23–25 and the second-order algebraic diagramatic construction scheme (ADC(2)) for the polarization propagator. 26–28 Simultaneously, the interest for highly stable (pseudo)spherical clusters has been revived in the last decade. Concretely, clusters that have closed-shell valence electron configuration according to the jellium model have been found to be particularly stable, receiving the generic name of “magic clusters”. 29,30 In particular, the Al− 13 anion and some of its dopped derivates have shown very promising properties. 31,32 Al− 13 is one of the most attractive magic clusters. It has a perfect icosahedral symmetry with an aluminum atom at the center, 33 a closed-shell electronic configuration (40 electrons), and a large highest occupied molecular orbital−lowest unoccupied molecular orbital (HOMO−LUMO) gap. 34 These features result in an unusual stability and chemical inertness, similar to noble gases, that make the clusters inert toward coalescence. But up to now their optical properties have reminded largely unexplored. We believe that these family of compounds are good candidates for the study of plasmonic-like resonances in small nanoclusters. The ultimate goal of this work is to explore and characterize collective excitations in the Al− 13 anionic cluster, and explore their plasmonic nature. In this article we visit two different approaches to identify and characterize the collectivity of electronic transitions, that is the use of a collectivity index as a measure of the number of electron/hole pairs participating in the excitation, and through the scaling of the electron-electron interactions. We apply these ideas to the study of simple models and to perform a deep analysis of the electronic excited states of Al− 13 . The present paper is organized as follows. First we discuss the two mentioned computational procedures for the characterization of collective excitations in molecules and nanoclusters. Then we study low energy electronic states of the Al− 13 cluster and explore higher energy optical plasmonic transitions. Finally, the main lessons derived from our study are highlighted in the Conclusions. All calculations presented in this paper have been performed with the Q-Chem program. 35

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Natural bond orbital (NBO) analysis was performed with the NBO 5.0 package. 36

Characterization of plasmonic transitions In the following, we present and discuss two different computational tools for the study of plasmon-like excitations in molecules and nanostructures.

Transition inverse participation ratio The transition density matrix between electron states Ψ0 and Ψn is defined as 0n Tpq = hΨ0 |p† q|Ψn i

(1)

where p† and q are creation and annihilation operators corresponding to spin orbitals φp and φq , respectively. This matrix (T) compresses the information contained in the many-body wave functions of the initial and final states in terms of the electronic transition. It can be used to compute interstate physical properties related to a one-particle operator Aˆ as expectation values: ˆ n i = Tr[AT] hΨ0 |A|Ψ

(2)

where Tr[AT] indicates the trace of the matrix product. In addition, it is also possible to employ operators not linked to observables, but related to physical and chemical concepts instead, in order to analyze the nature of the electronic transitions. Such ideas have been recently used in the description of spatial distributions and correlation effects of exciton wave functions. 37,38 The dimension of the T matrix is O ×V , where O and V denote the number of occupied and virtual molecular orbitals (MOs), respectively. The natural transition orbitals (NTOs) are constructed through a singular value decomposition (SVD) of the transition matrix T 39 as: T = UΛVT 5


(3)


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where Λ is a diagonal matrix with diagonal elements λi corresponding to transition occupation numbers, U is the unitary transformation matrices from canonical occupied MOs to a set of NTOs representing the excitation hole, V is a transformation matrix from the canonical virtual MOs to the NTOs representing the excited electron. Because the NTOs indicate the minimal necessary electron-hole pairs to describe excitations, it seems adequate to use them as the building pieces to define a measure for the collectivity of electronic transitions. On the other hand, the inverse participation ratio (IPR) is a mathematical tool that, in the frame of quantum mechanics, is employed to characterize the nature of electronic states. 40 The IPR value is typically used as an indicator of how many states a particle is distributed over, and it is obtained by applying a simple mathematical expression to the set of electron occupancies of the natural orbitals obtained as eigenvectors of the one particle density matrix for a system with nocc electrons. Here, we extend the use of the IPR expression to the analysis of electronic transitions by using the transition occupation numbers λi , and we name it transition inverse participation ratio (TIPR), labelled here as τ (Eq. 4).

τ=

"n occ X i=1

λ4i

#−1

(4)

The lower limit of the τ index is τ = 1, which corresponds to a unique hole to particle excitation, and the maximum excitation collectivity limit τ = nocc , in the case of λi = −1/2

nocc , ∀ i. These properties make the τ index a suitable quantity for the measure of the collectivity of electronic transitions. We would like to stress here that TIPR is a direct measure of the excitation’s collectivity, which is a necessary but not sufficient condition for plasmonic transitions. Thus, although τ can definitively be used to measure the number of electron-hole pairs involved in the transition between two electronic states, it cannot univocally identify plasmons. To get a clear view of the type of information that we can obtain from the τ index, we look to the nature of the lowest excited singlet state in He2 as a function of interatomic separation

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At approximately 5.2 Å interatomic distance τ of S1 abruptly drops to one electron-hole pair.

Scaling of electron-electron interactions Bernadotte and collaborators have recently proposed a systematic procedure to identify plasmonic excitations in molecular systems. 41 Their idea is based on the different dependence of single-particle excitations and plasmons on the Coulomb kernel. If plasmon-like excitations are defined as those corresponding to zeros of the dielectric function, there is a linear relation between the squared excitation energy of plasmons and the Coulomb interaction. On the other hand, single-particle transitions are expected to show much weaker dependence with electron-electron interactions. From this relationship, the authors analyze the dependence of the transition energy with the scaling of the electron-electron interaction in the response calculation within the TDDFT scheme. They chose to uniformly scale both Coulomb and exchange-correlation kernels arguing that they both emerge from the electron-electron interactions. On the other hand, the exchange contributions have no counterpart within the classical derivation of plasmons. Moreover, the presence of exact exchange in the linear response has a strong influence to non-local excitations, irrespectively of the collectivity of the transition. In this sense, the scaling of the exchange response in TDHF or in TDDFT with hybrid functionals would be probably more appropriate for the identification of charge transfer or Rydberg states. For these reasons, here we use the scaling approach by considering a scaling parameter γ acting only on the Coulomb kernel in the TDHF and TDDFT equations (Eq. 5), as it has been done recently for the configuration interaction singles and the second-order algebraic diagramatic construction scheme of the polarization propagator. 42 











(γ) 1 0  X  B(γ)   X   A (γ)    =ω    Y 0 −1 Y B∗(γ) A∗(γ)

8


(5)

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In Eq. 5 the matrix interaction elements of A and B matrices depend on the γ parameter as: (γ)

xc Aia,jb = (ǫa − ǫi )δij δab + γ(ia|jb) + fiajb (γ)

(6) (7)

xc Bia,jb = γ(ia|bj) + fiabj .

The terms in the right hand side of Eq. 6 are, from left to right, the energy difference between orbitals i and a, the two electron exchange integral (ia|jb) given in the Mulliken notation (Eq. 8), which corresponds to the response of the Coulomb operator, and the response of the exchange-correlation potential corresponding to a Coulomb like integral within the random phase approximation (Eq. 9) and the second functional derivative of the exchange-correlation energy, i.e. the exchange-correlation kernel, in TDDFT (Eq. 10).

(ia|jb) =

Z

d3 rd3 r′ φ∗i (r)φa (r)

1 φ∗j (r′ )φb (r′ ) ′ |r − r |

(9)

xc fiajb (RPA) = −(ij|ab) xc fiajb (TDDFT)

= (ia|fxc |jb) =

Z

d3 rd3 r′ φ∗i (r)φa (r)

(8)

δ 2 Exc φ∗ (r′ )φb (r′ ) δρ(r)δρ(r′ ) j

(10)

In order to test this scaling approach we explore the properties of electronic excited states in a small sodium cluster, that is the Na6 ring, which has been claimed to hold plasmon-like electronic transitions. 10 We compute the 50 lowest excited singlet states at the BP86/631G(d) level for the Na6 ring with D6h symmetry and taking 3.72 Å as the interatomic distance. The TIPR values identify some excited states as highly collective excitations. In particular there are three excited singlets belonging to the E2g , A2u and A1g irreducible representations with ∼ 6 electrons participating to the excitation, i.e. ∼ 3 electron/hole pairs (Figure 2).

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Figure 3: Transition energies to the 50 lowest excited singlet states of Na6 ring computed with different scaling factors (γ) at the BP86/6-31G(d) level. Selected states in Figure 2 are indicated in orange full cercles and continuous lines. Dashed grey lines only follow the computed state ordering. At this point it is important to emphasize that the TIPR quantifies the collective nature of electronic transitions, but it cannot be considered as a direct measurement of the optical response of the system to the action of an electromagnetic field. The above characterization of electronic excitations in the Na6 ring clearly illustrates this fact. Amongst the three strongly collective excitations identified, only one of them, the 1 A2u computed at 3.2 eV, shows strong optical activity (oscillator strength of 2.66), while the E2g and A1g collective transitions are dipole forbidden by symmetry. In this sense, the τ index introduced here cannot be directly correlated to other computational measurements that try to account for the optical response of the induced density, such as the plasmonicity index. 43 A comparison 11



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between TIPR and the plasmonicity index can be found in the Supporting Information.

Collective excitations in the Al− 13 cluster In the following, we explore the properties of electronic excited states of the Al− 13 cluster. In addition to the computation of excitation energies, transition dipole moments and symmetry characters, we also apply the computational tools described in the previous sections to identify and characterize the presence of plasmon-like transitions. The ground state geometry of the Al− 13 anionic cluster was optimized within the density functional theory (DFT) 44,45 using the PBE0 functional and the def2-TZVPD basis set. 32 This structure was subsequently employed to compute the electronic structure of singlet excited states within the TDDFT and with the Tamm-Dancoff approximation. 46,47 All values presented here have been obtained with the PBE exchange-correlation functional 48 in combination with the 6-31+G(d) basis set. Comparison of the performance of this approach to a large variety of functionals (with and without the TDA) and to time-dependent HartreeFock (TDHF), 49,50 shows only small quantitative differences between the results obtained at different levels. Dependence of excitation energies and oscillator strengths with exchangecorrelation functional and the basis set can be found in the Supporting Information (Tables S1 and S2). The 1400 lowest singlet-to-singlet transition energies and oscillator strengths (with no symmetry restrictions) of the Al− 13 cluster were considered in the simulation of the absorption spectrum. The profile of the absorption peaks were approximated by means of a Gaussian model with a full half-bandwidth of 3000 cm−1 .

Low-lying excited states of Al− 13 The Al− 13 anionic cluster presents an icosahedral symmetry with one aluminum atom at the center of the cluster and the other twelve atoms forming a cage around it, i.e. (Al@Al12 )− . Geometry optimization of Al− 13 indicates that all atoms on the surface are equidistant to the 12


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central atom at the interatomic distance of 2.661 Å and with slightly longer surface bonds between neighbor atoms (2.798 Å), in rather good agreement with previous studies. 33 The Al− 13 cluster has a closed shell electronic structure (Figure 4), with a relatively large energy gap between the four degenerate highest occupied orbitals (gu ), and the lowest unoccupied orbitals belonging to the hg symmetry, which is responsible for the strong stability of the Al− 13 cluster. It is worth noticing that there are two sets of triply degenerate occupied orbitals with energies rather close to the gu level, the t1u and t2u orbitals, while the virtual orbitals above hg are well separated in energy. As it has been pointed out in the literature, 51 the one-particle energy levels of Al− 13 follow, to a first approximation, the jellium model for spherical clusters 52,53 with 40 electrons. As a consequence of its high symmetry degree, the mono-electronic energy spectrum shows a rather high density of states at the frontier between occupied and unoccupied levels (Figure 4). Therefore, intuitively, it seems plausible to expect the presence of low-lying states obtained as the contribution of several electron/hole pairs, hence transitions with important collective character.

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This prediction is well recovered by our TDDFT calculations, with a large amount of states within a small excitation energy window (Table 1). The first 29 lowest excited states (11 Hu , 21 Hu , 11 Gu , 31 Hu , 11 T2u , 21 Gu 11 T1u ) are only spread over 0.1 eV, that is between 1.9 and 2.0 eV. The 11 T2u is computed at 2.1 eV and the 41 Hu , 41 Gu and 21 T1u set at 2.2 eV, while transition energies to the third T1u and T2u states are quite higher, i.e. about 2.9 eV. The next computed singlet state above the 31 T1u level appears at ∼0.5 eV higher in energy. It is worth mentioning that all these states energetically lie well below the vertical ionization potential, computed at 5.36 eV.

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Table 1: Transition energies ∆E (in eV) with oscillator strengths f (in parenthesis), charge increase at the central atom with respect to the ground state, τ index and main contributions to the excited state composition (in %) computed at the PBE/6-31+G(d) level.a

state

∆E (f )

11 Hu 21 Hu

∆q

τ

1.92 1.93

-0.11 -0.06

2.37 2.72

1 1 Gu

1.94

-0.02

3.11

31 Hu

1.98

0.34

2.24

11 T2u

1.99

-0.03

3.68

2 1 Gu

2.00

0.31

2.69

11 T1u

2.02 (0.020)

0.16

3.67

21 T2u

2.11

0.17

3.66

41 Hu

2.21

-0.01

2.02

3 1 Gu 21 T1u

2.22 2.23 (0.003)

-0.02 0.06

2.30 4.37

31 T2u

2.88

0.16

5.16

31 T1u

2.93 (0.106)

0.04

6.21

a Vertical

Composition gu → hg gu → hg t1u → hg gu → hg t1u → hg t1u → hg gu → hg gu → hg t1u → hg t2u → hg t1u → hg gu → hg gu → hg t1u → hg t2u → hg t2u → hg t1u → hg gu → hg t2u → hg gu → hg t2u → hg gu → hg t1u → hg t2u → hg gu → gg t1u → hg t1u → gg t2u → hg gu → gg t1u → hg gu → hg t1u → ag

ionization potential computed at 5.36 eV.

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Amongst the atomic-like molecular orbitals involved in the electronic transitions that define the low-lying states of Al− 13 (Figure 4), only the occupied t1u set exhibit sizable participations of the central aluminum, which indicates that electronic structure changes due to excitation mainly occur on the surface of the cluster, unless electrons are significantly promoted from the three t1u orbitals. This is also indicated by comparison between NBO analysis of the ground and excited states. In the ground state, the central aluminum atom holds the negative charge of the cluster, with a natural charge of -1.58e, while the atoms at the surface have slightly positive charges, that is 0.05e each. The differential charge at the central aluminum upon excitation, defined as the difference between the charge of the central atom at the ground and excited states (∆q = q(ES) - q(GS)), is very small (|∆q| ≤ 0.1e) for those states with minor contributions of excitations from t1u (Table 1). On the other hand, 31 Hu and 21 Gu states are mainly built as t1u → hg excitations and present the largest electron migration form the central aluminum, with ∆q = 0.34e and 0.31e, respectively. The 11 T1u and the two highest 1 T2u states correspond to intermediate cases, with ∆q = 0.16 − 0.17e. Amongst all different symmetry states in Al− 13 , those belonging to the triply degenerated T1u representation are the only dipole allowed transitions. Hence, these states will define the profile of the absorption spectrum and the light-harvesting properties of the cluster. The transition energy to the lowest optical transition is computed at 2.02 eV, while the 21 T1u state appears ∼ 0.2 eV higher and it is predicted to have a weaker oscillation strength. The 31 T1u state at ∆E = 2.93 eV presents the largest transition probability, and it is predicted to originate the main feature of the absorption spectrum at low energies.

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Figure 7: Change of the squared normalized excitation energies of the Al− 13 cluster computed at the PBE/6-31+G(d) with the scaling factor γ.

Optical plasmons of Al− 13 As discussed in previous sections, the participation of multiple electron/hole pairs in electronic excitations, that is the collective degree of the electronic transition as measured by the TIPR, does not univocally imply strong optical response to an external electromagnetic field. In particular, the highly collective low lying states of Al− 13 (Table 1) show rather small responses, with at most moderate computed oscillator strengths (f ≤ 0.1). On the other hand, we expect optical plasmons with intense transitions to exhibit large TIPR values. Hence, to identify collective optical excitations in Al− 13 we simulate the absorption spectrum for a wide range of transition energies (Figure 8).

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that is with the effective participation of 14 electrons, corresponding to one electron per Al atom plus the additional electron of the anionic Al− 13 cluster. The most intense feature in the simulated spectrum can be mainly assigned to the excitation computed at 10.5 eV (f = 3.11). The main contributions to this state correspond to electron promotions from hg and t1u occupied orbitals (Figure 4) to the t2u and hg virtual orbitals, corresponding to D and F like orbitals of the jellium model. The TIPR value for this transition also suggests important collective character, with τ = 7.1, just slightly larger than the value obtained for the 8.5 eV transition. Although the analyzed states related to the main features of the absorption spectra of Al− 13 cluster exhibit considerably large values of τ , it does not exclude the possibility of the existence of other highly collective transitions within the explored excitation energy range with low or no optical activity. In other words, large τ values do not univocally imply the presence of optical plasmons. This is shown in Figure 9 (left), where we represent the collectivity index as a function of the excitation energy. Although it is true that at higher energies it is possible to find transitions with higher TIPR values, we find highly collective excitations independently of the transition energy.

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the TIPR to the study of plasmons in the Al− 13 cluster in combination with the scaling of electron-electron interaction, which has been successfully applied to the study of plasmons in molecules in several occasions. The electronic structure of low-lying states of Al− 13 has been obtained with TDDFT. The icosahedral symmetry of the cluster induces a large density of states obtained as electronic excitations from the highest occupied to the lowest unoccupied energy levels. In our analysis we have identified the existence of low-lying electronic states with strong collective character in Al− 13 , which might be labelled as plasmon-like transitions. The NBO analysis for the ground and excited states characterize most of the computed transitions as delocalized on the cluster surface, with rather small participation of the electrons on the central aluminum. Therefore, the low energy collective excitations in Al− 13 might be characterized as surface plasmon-like transitions. High collectivity of the transition does not imply strong response to the action of an electromagnetic field. In this sense, to identify optical plasmons in nanostructures can be achieved by combining TIPR values with computed oscillator strengths. Our results suggest that collective optical transitions in Al− 13 are reached at higher frequencies. These are encouraging results, which motivate further studies of collective excitations in aluminum clusters and in other related compounds, such as doped clusters derived from Al− 13 . We expect that the application of the TIPR index to other clusters and molecular systems will help to validate it as a collectivity measure of electronic excitations, and will prove its usefulness in the study of plasmonic excitations in finite systems. These works are currently under development in our laboratory.

Acknowledgement The authors are grateful to Professor Vladimiro Mujica for his valuable insights. This work has been supported by the Basque Government (project IT588-13) and the Spanish Government (project CTQ2012-38496-C05-04). The authors gratefully acknowledge SGIker for allocation of computational resources. D.C. acknowledges IKERBASQUE, Basque Founda24


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tion for Science for financial support.

Supporting Information Available The following files are available free of charge. • casanova_SI.pdf: contains a comparison of the TIPR indicator with the plasmonicity index in the computation of optical excitations of naphthalene, tables with the dependence of excitation energies and oscillator strengths with exchange-correlation functional and basis set, and the xyz coordinates for the optimized geometry of the Al− 13 cluster.

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