Quantifying plasmonic characteristics of a series of pure Aluminum and alkali doped Aluminum clusters

doi:10.21203/rs.3.rs-1974642/v1

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Quantifying plasmonic characteristics of a series of pure Aluminum and alkali doped Aluminum clusters

https://doi.org/10.21203/rs.3.rs-1974642/v1

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Al clusters have drawn tremendous attention of scientific community for their characteristic deep ultraviolet plasmonic emission. In this study, we have explored the plasmonic characteristics of a series of linear (Al_n; n = 2 to 9) and cyclic aluminum clusters (n = 3,4) along with our recently reported three Al₁₃⁺ isomer system [Guin et. al. Journal of Molecular Graphics and Modeling, 2020, 97, 107544] and corresponding alkali doped clusters [Guin et. al. Journal of Molecular Modeling, 2021, 27, 235]. Among the three Al₁₃⁺ isomers one is perfectly planar (CI) and two others are quasi-planar clusters (CT and CII). It is a well-known fact that properties of nano-clusters strongly depend on the size and shape of the clusters. The current study reveals that for the linear chains the plamonic character systematically increases with the nuclearity of the clusters. For the cyclic clusters (Al₃ and Al₄) the plamonicity is lower compared to corresponding linear clusters. In case of Al₁₃⁺ isomer system, the planar isomer is having higher plasmonicity compared to the non-planar isomers. The doping of alkali atoms in planar clusters, has a general tendency to reduce the plasmonic nature compared to the undoped clusters. The plasmonic emission for these aluminum clusters fall in the UV range (170 to 416 nm).

Linear Al chain

Triangular and square shaped Al cluster

Pure planar/quasi-planar Al13+ cluster

Alkali doped Al13+ cluster

TDDFT study

Canonical Molecular Orbital

HOMO and LUMO

Polarizability

Density of states

Utilization of plasmonic emission of metal nano clusters is currently revolutionizing diverse areas of nano technology, such as chemical [1] and biological [2] sensing even at single molecule level, plasmon driven enhancement of solar cell efficiently [3], plasmon driven controlled chemical reaction [4] and multifold enhancement of catalytic activity [5], metamaterial with exotic physical phenomena like negative refractive index and cloaking [6] etc. Though plasmonic emission has been traditionally observed for metal clusters such as gold [7], silver [8], Aluminum [9], copper [10] etc., recently plasmonic emission of molecules [11] also have drawn considerable attention. Molecular plasmonic emission as well as the plamonic emission of metal nano-clusters [12] with size < 2nm are somewhat analogous as both belong to the area of quantum plasmonics [13]. There is still an ongoing evolution in the ideas of how to analyze and characterize the plasmonic characteristics of molecules and very small nano clusters [14].

Among metal nano clusters, Gold has now become the standard substrate for SERS detection due to its plasmonic emission in the visible spectral range. Silver nano-clusters are also widely employed in various areas of applications [15], but plasmonic properties of Al nano-clusters have drawn special attention [16] as the plasmonic band of Al fall in the UV region, unlike gold and silver. UV plasmonic emission of Aluminum specialized applications like, Al Plasmon nano-thermometry through temperature assesement in single plasmonic hot spot [17] for which other metal nano clusters are not suitable. Also, Aluminum being a cheaper material commercial utilization of Aluminum based plasmonic devices has a greater potential. Plasmonic properties of metal nanoclusters depend both on the size and shape of the clusters as well as other environmental factors such as applied electric field [18]. The aim of the present study is to understand this effect on the plamonic emission characteristics of aluminum clusters. We have chosen the linear chains of aluminum atoms containing two to nine atoms and cyclic Al₃ and Al₄ clusters. Besides we have extended the study to our recently reported a rare set of three isomers of Al₁₃⁺ cluster one of which is perfectly planar (CI) and two others are quasi-planar (CT and CII) in natures. Among the two quasi-planar clusters CII is the global minimum energy with its flexible tail region disposed upwards whereas that of the other quasiplanar cluster CT is configured downward of the mean plane of the clusters [19,20]. Like poyacene molecules [21] which show interesting plasmonic property due to their characteristic aromatic nature, these three Aluminum isomers have also strong aromatic regions interspersed with anti-aromatic regions indicating a probable plasmonic character of the present Al₁₃⁺ system. It was quite inquisitive thus to understand the plasmonic properties of these planar, true minima quasi-planar Al₁₃⁺ isomers and their alkali doped structure. We have performed an in depth TDDFT analysis to characterize the low lying excited states of the threes pure and doped structures, looked for the possible plamonic states and the nature of the plasmonic transitions. It reveals that for the linear chains the plamonic character systematically increases with the number of aluminum atoms constituting the cluster. For the cyclic clusters (Al₃ and Al₄) the plamonicity is lower compared to corresponding linear clusters. Among the three Al₁₃⁺ isomers, plasmonicity character are in the order, planar Al₁₃⁺ cluster(CI) > Transition state Al₁₃⁺ cluster(CT) > True minima Al₁₃⁺ cluster(CII). The doping of alkali atoms in planar clusters reduce the plasmonic nature compared to the undoped clusters. This study reveals that plasmonic emission frequency which falls in UV range (170 to 415 nm) depends on the shape and size of the aluminum clusters, as well as on the nature of doping.

Tddft And Plamonicity:

Time-dependent density-functional theory (TDDFT) is a quantum mechanical theory used in physics and chemistry to investigate the properties and dynamics of many-body systems in the presence of time-dependent potentials. The effect of such fields on molecules and solids can be studied with TDDFT to extract features like excitation energies, frequency-dependent response properties, and photo absorption spectra. TDDFT is an extension of density-functional theory (DFT), and the conceptual and computational foundations are analogous – to show that the (time-dependent) wave function is equivalent to the (time-dependent) electronic density, and then to derive the effective potential of a fictitious non-interacting system which returns the same density as any given interacting system. The issue of constructing such a system is more complex for TDDFT, most notably because the time-dependent effective potential at any given instant depends on the value of the density at all previous times. Consequently, the development of time-dependent approximations for the implementation of TDDFT is behind that of DFT, with applications routinely ignoring this memory requirement. Here we have studied the effect of TDDFT to observe the plasmonic effect of our reported Al₁₃⁺ clusters.

The electronic states among which have greater Transition Dipole Moment (TDM) or oscillator strength is called Bright Plasmon and having least Transitiosn Dipole Moment (TDM) or oscillator strength is called Dark Plasmon [22].

TDDFT [23] computations have been performed using for the pure as well as doped clusters using Gaussian 03 package [24] employing 6-311 + G(d) basis set [25] and B3LYP hybrid density functional [26]. Canonical molecular orbitals were generated using Chemissian package [27]

A brief introduction of the studied system:

Optimized geometry of small Al clusters

A series of Aluminum chain containing 2 to 9 atoms along with a triangular and a square shaped small clusters have been depicted in Fig. 1. All of these clusters are optimized structure with zero or more imaginary frequencies. For Al2 cluster interatomic distance is 2.531 Å, it is 2.472 Å for Al3 cluster, 2.728 Å for Al4 cluster, 2.734 Å for Al5 cluster, 2.738 Å for Al6 cluster, 2.693 Å for Al7 cluster, 2.723 Å for Al8 cluster, 2.721 Å for Al9 cluster. For triangular Al3 cluster, it is 2.509 Å and for square shaped Al4 cluster, it is 2.600 Å. Length of the Al chain with with number of atom is proportional which has been plotted in Fig. 2.

Now the turn to discuss the set of planar and quasi-planar Al₁₃⁺ cluster whose optimized structure and electronic character have already been reported by us [19, 20]. Among these three clusters one is perfectly planar whereas two other are quasi-planar structures. Among these two quasi- At these structures, one atom has been replaced from any symmetry position by alkali atoms (Li, Na and K) [28]. Among, all these set of structures, doping at position 4 are the minimum energy structures which are depicted in Fig. S1.These structures have been chosen to see the plasmonic effect.

Time Dependent Density Functional Theory (Tddft):

TDDFT study for small Al clusters

TDDFT calculations provide a nice idea about TDM and OS of the structures. For, pure linear Al₂ cluster, bright plasmons have been found at state 11 with TDM value and Oscillator strength are − 2.79 debye and 0.87 respectively (see Table 1).For, Al₃ cluster this TDM and oscillator strength have been increased significantly to 4.75 Debye and 2.23 respectively (State 28). For, Al₄ cluster, bright plasmons have been found at state 17 with TDM value and Oscillator strength are − 6.72 Debye and 1.95 respectively. In case of Al₅ cluster, TDM and oscillator strength have also been enhanced significantly to -6.19 Debye and 3.31 respectively (State 55). For, Al₆ cluster, bright plasmons have been found at state 33 with TDM value and Oscillator strength are − 6.89 Debye and 3.97 respectively. For, Al₇ cluster this TDM and oscillator strength have been increased significantly to -7.48 Debye and 4.40 respectively (State 28). For, Al₄ cluster, bright plasmons have been found at state 17 with TDM value and Oscillator strength are − 6.72 Debye and 1.95 respectively. Whereas, for Al₅ this TDM and oscillator strength have also been enhanced to -6.19 debye and 3.31 respectively (State 55). So, there is a systematic variation of OS and TDM with Al chain has been plotted in Fig. 2.

Table 1

Table for highest plasmonics transition of small Al clusters obtained from TDDFT calculation.
Cluster	Length of the chain	Excited State	Optimized energy in Hartree	Absorbed wavelength in nm	Oscillator strength	TDM
Linear Chain
Al₂	2.531	11	-484.5209875	270.54	0.87	-2.79
Al₃	4.943	28	-726.8232394	307.94	2.23	4.75
Al₄	7.980	17	-969.1440582	348.10	1.95	-4.72
Al₅	10.371	55	-1211.4548099	351.94	3.31	-6.19
Al₆	12.784	33	-1453.7481465	363.42	3.97	-6.89
Al₇	15.245	80	-1696.0604769	386.67	4.40	-7.48
Al₈	17.636	49	-1938.344392	395.60	4.37	7.54
Al₉	20.057	114	-2180.6589412	415.83	4.83	-8.13
Triangle
Al₃	2.509	73	-726.880578	182.43	1.14	6.8180
Square
Al₄	2.600	74	-969.186197	173.90	0.98	5.5902

Tddft Study For Undoped And Doped Al Clusters

The TDDFT study reveals that for the undoped planar Al₁₃⁺ cluster the bright plasmonic state is the state 189 with TDM value 4.37 Debye and Oscillator strength 2.59. Details of this transition have been jotted down in Table 2 and Table 3. For quasi-planar cluster, this TDM and oscillator strength has been reduced significantly to 2.27 Debye and 0.68 respectively which is observed for the state 185. Alkali atom doping has also significant effect on the TDM and OS and generally the TDM and OS decreases upon doping. For the Li doped planar Al₁₃⁺ cluster state 174 correspond to the bright plasmon state which has TDM value and Oscillator strength 3.07 Debye and 1.17 respectively. For Li doped quasi-planar cluster this TDM and oscillator strength have been reduced significantly to 1.17 Debye and 0.72 respectively (State 188). For, planar Na doped Al₁₃⁺ cluster, bright plasmons have been found at state 178 with TDM value and Oscillator strength are 2.65 Debye and 0.88 respectively. Similarly, for Na doped quasi-planar cluster this TDM and oscillator strength have been reduced significantly to 3.07 debye along Y-axis and 1.30 respectively (State 196). For, planar K doped Al₁₃⁺ cluster, bright plasmons have been found at state 179 with TDM value and Oscillator strength are 3.25 Debye and 1.29 respectively. But in case of K doped quasi-planar cluster this TDM and oscillator strength have been enhanced to 3.29 Debye along Y-axis (-0.46 Debye along z-axis due to quasi-planarity) and 1.70 respectively (State 202). Detail study of all plasmonic parameters for different clusters has been given in Table 4. Undoped Transition state Al₁₃⁺ cluster exhibits more plasmonic character than planar cluster but less plasmonic behaviour than quasiplanar cluster (Table S1). In each cases, major contribution of transition dipole moment has been found along Y-axis, in which bilateral symmetry axis of each cluster passes. But, in case of quasi-planar clusters these major contribution of TDM value along Y- axis is shared towards Z-axis due to non-planarity of the cluster. Naturally, dipole moment along a specific direction (Y-axis/ symmetry axis) has been reduced significantly, in case of corresponding quasi planar clusters. Oscillator strength which is proportional to TDM is also reduces, it indicate plasmonicity also reduces.

Table 2

Table for highest plasmonics transition of planar Al₁₃⁺ clusters obtained from TDDFT calculation.
Clusters	State	Transition Energy ∆E in eV	Transition (Contributions)	∆n	Transition Weight	Electric Transition Dipole Moments (TDM)			Oscillator Strength(f)
Clusters	State	Transition Energy ∆E in eV	Transition (Contributions)	∆n	Transition Weight	µ_x	µ_y	µ_z	Oscillator Strength(f)
Planar Clusters
Pure Al₁₃⁺ Cluster	189	5.5253	H -> L + 25 (21%)	26	0.32205	0.0004	4.3749	0.0000	2.5909
			H-5-> L + 14 (11%)	20	0.23239
			H-12 -> L (7%)	13	0.18053
			H-3 ->L + 20 (7%)	24	0.18526
			H-3 ->L + 23 (6%)	27	0.17912
			H-5 ->L + 19 (4%)	25	-0.13323
			H-7 -> L + 10 (4%)	18	-0.13308
			H-2 ->L + 22 (3%)	25	-0.12511
			H-5 ->L + 17 (3%)	23	0.11841
			H-2 ->L + 19 (2%)	22	-0.10041
Li doping	174	5.0860	H-8->L + 6 (11%)	15	0.23204	3.0729	0.0020	0.0000	1.17
			H-7->L + 12 (25%)	20	0.11406
			H-6->L + 14 (16%)	21	0.35035
			H-8->L + 8 (3%)	17	-0.15449
			H-6->L + 10 (5%)	17	0.27844
			H-6->L + 16 (5%)	23	-0.15522
			H-3->L + 21 (3%)	25	-0.12555
			H->L + 26 (8%)	27	0.20037
Na doping	178	5.0972	H-7->L + 12 (23%)	20	0.33977	0.0001	2.6508	0.0000	0.88
			H-6->L + 14 (11%)	21	0.23146
			H-6->L + 16 (34%)	16	0.41436
			H-3->L + 21 (4%)	16	-0.14426
			H->L + 25 (4%)	26	-0.14494
K doping	179	4.9901	H-8->L + 6(14%)	15	0.26760	0.0002	3.2478	0.0000	1.29
			H-6->L + 16 (25%)	23	0.35385
			H-7->L + 14 (6%)	22	-0.17591
			H-6->L + 11 (4%)	18	-0.14795
			H-6->L + 18 (3%)	25	0.13107
			H-4->L + 17 (5%)	22	0.15866
			H-3->L + 10 (2%)	14	-0.10910
			H-3->L + 21 (6%)	25	0.16746
			H-2->L + 18 (2%)	21	-0.10478
			H->L + 27 (4%)	28	-0.13594
			H->L + 35 (4%)	36	0.14602

Table 3

Table for highest plasmonics transition of quasi-planar Al₁₃⁺ clusters obtained from TDDFT calculation.
Clusters	State	Transition Energy ∆E in eV	Transition (Contributions)	∆n	Transition Weight	Electric Transition Dipole Moments (TDM)			Oscillator Strength(f)
Clusters	State	Transition Energy ∆E in eV	Transition (Contributions)	∆n	Transition Weight	µ_x	µ_y	µ_z	Oscillator Strength(f)
Quasi-planar Clusters
Pure Al₁₃⁺ Cluster	185	5.3768	H-1->L + 22 (18%)	24	0.30168	2.2707	-0.0011	-0.0668	0.68
			H-9 -> L + 7 (17%)	17	-0.29278
			H-8 -> L + 11 (9%)	20	-0.20618
			H-2 -> L + 21 (8%)	24	-0.19914
			H-2 ->L + 20 (5%)	23	0.15992
			H-5 ->L + 19 (4%)	25	0.14463
			H-3 ->L + 19 (3%)	23	-0.12052
			H -> L + 23 (3%)	24	0.12147
Li doping	188	5.6113	H-4->L + 20 (31%)	25	-0.10528	0.0169	-2.3003	-0.0038	0.72
			H-11->L + 5 (7%)	17	0.19042
			H-7->L + 13 (6%)	21	0.17952
			H-6-> L + 15 (2%)	22	-0.10981
			H-6-> L + 17 (9%)	24	0.21201
			H-5-> L + 18 (4%)	24	-0.14332
			H-5-> L + 20 (2%)	26	0.39438
			H-3-> L + 19 (3%)	23	0.12151
Na doping	196	5.6381	H-10->L + 7 (13%)	18	-0.25106	-3.0653	0.0004	0.0699	1.30
			H -2-> L + 21 (16%)	24	-0.21331
			H-7->L + 14 (9%)	22	-0.14356
			H-6->L + 20 (4%)	27	-0.11638
			H-5->L + 14 (3%)	20	0.11596
			H-5->L + 18 (3%)	24	-0.12056
			H-5->L + 21 (3%)	27	-0.12384
			H-4->L + 18 (3%)	23	0.27847
			H-1->L + 25 (3%)	27	-0.12663
			H-1->L + 26 (3%)	18	-0.11838
K doping	202	5.6072	H-11->L + 2 (3%)	14	-0.11799	3.4921	0.0000	-0.4644	1.70
			H-10->L + 7 (6%)	18	0.16918
			H-8 -> L + 8 (9%)	17	0.21429
			H-5-> L + 14 (4%)	20	0.14802
			H-5-> L + 18 (4%)	24	-0.13577
			H-5->L + 21 (5%)	27	0.16383
			H-4->L + 18 (2%)	23	0.10386
			H-1->L + 27 (8%)	29	-0.19779
			H-> L + 26 (4%)	27	-0.13618

Table 4

Variation of absorbed wavelength at resonance with dimension of the clusters.
Planar/ Quasi-planar	Pure/Doping	Dimensions			States at maximum oscillator strength	Absorbed wavelength at resonance in nm	Absorbed energy at resonance in eV	Oscillator strength
Planar/ Quasi-planar	Pure/Doping	Max horizontal length in Å	Max vertical length in Å	Average Length in Å	States at maximum oscillator strength	Absorbed wavelength at resonance in nm	Absorbed energy at resonance in eV	Oscillator strength
Planar	Pure Al₁₃⁺ Cluster	7.95	7.53	7.74	189	224.39	5.53	2.59
	Li doping	7.83	8.96	8.39	174	243.78	5.09	1.17
	Na doping	7.83	9.38	8.60	178	243.24	5.10	0.88
	K doping	7.81	10.03	8.92	179	248.46	4.99	1.29
Quasi-planar	Pure Al₁₃⁺ Cluster	7.51	7.36	7.43	185	230.59	5.38	0.68
	Li doping	6.92	6.68	6.80	188	220.95	5.61	0.73
	Na doping	7.08	6.69	6.88	196	219.90	5.63	1.30
	K doping	7.11	6.72	6.91	202	221.11	5.60	1.70

Uv-vis Spectra Study:

UV-Vis spectra for small Al clusters

UV-Vis spectra [29] of these three clusters have been depicted in Fig. 3. For linear Al₂ cluster three consecutive distinct sharp resonance peaks have been observed, among them peak located at 270.54 nm with oscillator strength 0.87 is highest. Whereas for Al₃ cluster, it is located at 307.94 nm which exhibits significant oscillator strength of 2.23. For, Al₄ chain, a sharp resonance peak has been observed at 348.10 nm with oscillator strength 1.95. Whereas, for Al₅ cluster, it is located at 351.94 nm with oscillator strength 3.31. For Al₆ chain, this sharp resonance peak has been observed at 363.42 nm with oscillator strength 3.97. Whereas, for Al₇ cluster, it is located at 386.67 nm with oscillator strength 4.40. For Al₈ cluster, this sharp resonance peak has been observed at 395.60 nm with oscillator strength 4.97. Whereas, for Al₉ cluster, it is found at 415.83 nm having oscillator strength 4.83. Now turns for 2d clusters. For triangular Al₃ cluster, highest plasmonic emission is observed at 182.43 nm with oscillator strength 1.14. For square shaped Al₄ cluster, it is found at 173.90 nm with oscillator strength 0.98. So, resonance absorbance coefficient which is directly proportional to number of plasmons associated with this resonance or oscillator strength is higher for planar clusters than its corresponding quasi-planar cluster. Also, resonance frequency is higher for planar clusters than its corresponding quasi-planar clusters. So, for these pure and doped Aluminum clusters, the plasmon band is located in the ultraviolet region. Generally, for usually available Ag, Au metal nanoparticles, it is seen within IR region. So, resonance absorbance frequency of Al nanoparticle is high. Also, this resonance absorption frequency can be finely tuned by distorting and doping the clusters.

UV-Vis spectra for Al ₁₃ ⁺ clusters.

UV-Vis spectra of planar clusters and quasi-planar clusters have been depicted in Fig. 4. For pure planar Al₁₃⁺ cluster, sharp resonance peak has been observed at 224.39 nm with oscillator strength 2.59.Whereas, for undoped true minima quasi-planar Al₁₃⁺ cluster, it is located at 230.59 nm with oscillator strength 0.6798. For undoped transition state Al₁₃⁺ cluster, sharp resonance peak has been observed at 228.21 nm with oscillator strength 1.10 (Fig. S2). For planar Li doped Al₁₃⁺ cluster, this resonance peak has been observed at 243.78 nm with oscillator strength 1.17. Whereas, for quasi-planar Li doped Al₁₃⁺ cluster, it is located at 220.95 nm with oscillator strength 0.72. For planar Na doped Al₁₃⁺cluster, this sharp resonance peak has been observed at 243.24 nm with oscillator strength 0.88. Whereas, for quasi-planar Na doped Al₁₃⁺ cluster, it is located at 219.90 nm with oscillator strength 1.30. For planar K doped Al₁₃⁺cluster, a sharp resonance peak has been observed at 229.70 nm with oscillator strength 1.8362. Whereas, for quasi-planar K doped Al₁₃⁺ clusters, it is located at 221.11 nm with oscillator strength 1.70. On the other hand, resonance absorbance coefficient which is directly proportional to number of plasmons associated with this resonance is higher for planar clusters than its corresponding quasi-planar clusters. Also, resonance frequency is higher for planar clusters than its corresponding quasi-planar clusters. So, for these pure and doped Aluminum clusters, the plasmon band is located in the ultraviolet region. Generally, for usually available Ag, Au metal nanoparticles, it is seen within IR region. So, resonance absorbance frequency of Al nanoparticle is high. Also, this resonance absorption frequency can be finely tuned by restructuring and doping the clusters.

Canonical Molecular Orbitals:

Canonical Molecular Orbital for small Al clusters

Canonical molecular orbitals [30] for Al chain with Triangular and square shaped Al clusters have been shown in Fig. 5. In case of Al₂ and Al₃, HOMO is delocalized due to \(\pi\) electrons but LUMO is localized due to\(\sigma\)electrons. In case of Al₄ system, both HOMO and LUMO are delocalized due to \(\pi\) electrons. In Al₅ cluster, \(\pi\) electron contributes to HOMO and it is delocalized, but is localized by \(\sigma\) electron. In case of Al₆ and Al₇ clusters, HOMO is delocalized due to \(\pi\) electron, whereas LUMO are localized due to \(\sigma\) electron. In case of Al₈ and Al₉ clusters, both localized HOMO and LUMO have contribution to \(\pi\) electron. In case of triangular Al₃ and Al₄ clusters, HOMO are delocalized and this comes due to \(\pi\) electron. LUMO is not delocalized and it has contribution to \(\sigma\) electron. So, most of the HOMO orbital are delocalized due to \(\pi\) electron may also the origin of exhibiting plasmonic character.

Canonical Molecular Orbital For Small Al Clusters

Canonical Molecular Orbital of planar and quasi-planar clusters has been shown in Fig. 6. Here, canonical molecular orbital of highest oscillator strength with greater transition weight have been depicted. For pure planar Al₁₃⁺ cluster, delocalized orbitals of transition from HOMO to LUMO + 25 have probability 21%. In this case, delocalized \(\pi\) orbitals have been confined in a shorter region after transition. Next higher transition from HOMO-5 to LUMO + 14 has probability 11%. In this case, σ delocalized orbitals have been confined in a shorter region after transition. For planar Li doped Al₁₃⁺ cluster, delocalized orbitals of transition from HOMO-7 to LUMO + 12 has probability 25%. In this case, contributions of both orbitals are due to \(\sigma\) electrons. Transition from HOMO-6 to LUMO + 16 have probability 11%. In this case, transition has been taken place from σ delocalized orbitals to another σ delocalized orbital. For planar Na doped Al₁₃⁺ cluster, delocalized orbitals of transition from HOMO-6 to LUMO + 16 has probability 34%. In this case, a small contribution of \(\pi\) orbitals is found after transition from delocalized σ orbital. Next higher transition from HOMO-7 to LUMO + 12 has probability 23%. In this both cases, contribution of delocalized σ orbital is found. For planar K doped Al₁₃⁺ cluster, delocalized orbital of transition from HOMO-6 to LUMO + 16 has probability 25%. In this case, transition occurs from delocalized σ orbital to \(\pi\) orbitals. Next higher transition from HOMO-8 to LUMO + 6 has probability 14%.Now the turns come for quasi-planar clusters. For pure quasi-planar Al₁₃⁺ cluster, delocalized orbitals of transition from HOMO-1 to LUMO + 22 has probability 18%. In this case, transition has been taken place from delocalized \(\pi\) orbitals to σ orbitals. Next higher transition from HOMO-9 to LUMO + 7 has probability 17%. In this case, area of σ delocalized orbitals have been confined in a shorter region after transition. For quasi-planar Li doped Al₁₃⁺ cluster, delocalized orbitals of transition from HOMO-4 to LUMO + 20 has probability 31%. In this case, transition has been taken from delocalized \(\pi\)orbitals \(\sigma\) orbitals.Transition from HOMO-6 to LUMO + 17 have probability 9%. In this case, delocalized \(\pi\) orbitals have been confined in a shorter region after transition from delocalized σ orbitals. For quasi-planar Na doped Al₁₃⁺ cluster, delocalized orbitals of transition from HOMO-2 to LUMO + 21 has probability 16%. In this case, transition occurs from delocalized \(\sigma\) orbitals to similar σ orbitals. Next higher transition from HOMO-10 to LUMO + 7 has probability 13%. In this case, delocalized \(\sigma\) orbitals have been confined in a shorter region after transition. For quasi-planar K doped Al₁₃⁺ cluster, delocalized orbitals of transition from HOMO-8 to LUMO + 8 have probability 9%. In this case, delocalized \(\sigma\) orbitals have been confined in a shorter region after transition. Next higher transition from HOMO-1 to LUMO + 27 has probability 8%. In this case, transition occurs from delocalized \(\sigma\) orbitals to similar σ orbitals. For undoped transition state Al₁₃⁺ cluster, delocalized orbitals of transition from HOMO-5 to LUMO + 19 have probability 23%. In this case, delocalized \(\sigma\) orbitals have been confined in a shorter region after transition. Next higher transition from HOMO-4 to LUMO + 18 has probability 17%. In this case, transition occurs from delocalized \(\sigma\) orbitals to another σ orbitals. So, most of the contribution of electronic transition which is generally plasmonic transition is found in delocalized orbitals.

Polarizability Study:

Polarizability is a measure of how easily an electron cloud is distorted by an electric field [31]. Generally, the electron cloud will being to an atom or molecule or ion. But, in this case electron cloud is due to plasmonic nanopaticle of Al₁₃⁺ nanoclusters. It can be represented in tensor or matix form for any arbitrary body. Sometimes due to symmetry of the system, its off-diagonal vanish. This is called anisotropic polarizability, it is represented by-

α_iso=\(\frac{1}{3}\) (α_xx + α_yy + α_zz)

This expression is applicable in our planar system.

However, during the dynamics, when the planar clusters transform into quasi-planar clusters, the nuclear displacements due to the activated bending mode cause a change in the polarizability that cannot be described in terms of its diagonal elements. The plane deviates from the planar symmetry and for this reason the off-diagonal polarizability contributions have to be included in the molecular polarizability. The anisotropic term, including the off-diagonal terms, can be represented by

α_aniso= \(\frac{1}{\sqrt{2}}\sqrt{{({{\alpha }}_{\text{x}\text{x}}-{{\alpha }}_{yy})}^{2}+{({{\alpha }}_{\text{x}\text{x}}-{{\alpha }}_{zz})}^{2}+{({{\alpha }}_{yy}- {{\alpha }}_{zz})}^{2}+6({{\alpha }}_{xy}^{2}+{{\alpha }}_{\text{y}\text{z}}^{2}+{{\alpha }}_{zx}^{2})}\)

Polarizability of these reported undoped and doped clusters has been provided in Table 5. For pure planar Al₁₃⁺ cluster, y-component of polarizability (α_yy )is largest (837.28 unit) and z-component of polarizabilty (α_zz ) is smallest (279.04 unit).There is no cross/off-diagonal components of polarizability. Isotropic and anisotropic polarizabilities are respectively 642.99 and 546.34 units. For pure quasi-planar Al₁₃⁺ cluster, y-component of polarizability (α_yy )is largest(788.16 unit) and z-component of polarizabilty (α_zz ) is smallest (332.14 unit).There also exist little amount of cross/off-diagonal components of polarizability due to quasi-planarity and principal component of polarizability reduces simultaneously. Isotropic and anisotropic polarizabilities are respectively 641.75 and 492.86 units. For undoped transition state Al₁₃⁺ cluster, y-component of polarizability (α_yy )is largest (802.84 unit) and z-component of polarizabilty (α_zz ) is smallest (310.02 unit). Like quasi-planar true minima cluster, there also exist little amount of cross/off-diagonal components of polarizability due to quasi-planarity and principal component of polarizability reduces simultaneously. Isotropic and anisotropic polarizabilities are respectively 645.19 and 507.98 units (Table S2). For planar Li doped Al₁₃⁺ cluster, y-component of polarizability (α_yy ) is largest (864.39 unit) and z-component of polarizabilty (α_zz ) is smallest (271.61 unit).There is no cross/off-diagonal components of polarizability. Isotropic and anisotropic polarizabilities are respectively 644.70 and 378.14 units. For Li doped quasi-planar Al₁₃⁺ cluster, y-component of polarizability (α_yy ) is also largest (677.51 unit) and z-component of polarizabilty (α_zz ) is smallest (364.02 unit).There also exist little amount of cross/off-diagonal components of polarizability due to quasi-planarity and eventually principal component of polarizability reduces simultaneously. Isotropic and anisotropic polarizabilities are respectively 568.58 and 400.74 units. For planar Na doped Al₁₃⁺ cluster, y-component of polarizability (α_yy) is largest (907.09 unit) and z-component of polarizabilty (α_zz ) is smallest (277.41 unit).There is no cross/off-diagonal components of polarizability. Isotropic and anisotropic polarizabilities are respectively 663.85 and 828.87 units. For Na doped quasi-planar Al₁₃⁺ cluster, y-component of polarizability (α_yy` ) is largest (693.96 unit) and z-component of polarizabilty (α_zz ) is smallest (370.51 unit).There also exist little amount of cross/off-diagonal components of polarizability due to quasi-planarity and eventually principal component of polarizability reduces simultaneously. Isotropic and anisotropic polarizabilities are respectively 580.00 and 256.29 units. For K doped planar Al₁₃⁺ cluster, y-component of polarizability (α_yy ) is largest (706.30 unit) and z-component of polarizabilty (α_zz ) is smallest (379.80 unit).There also exist little amount of cross/off-diagonal components of polarizability due to quasi-planarity and eventually principal component of polarizability reduces simultaneously. Isotropic and anisotropic polarizabilities are respectively 591.47 and 345.68 units. For K doped quasi-planar Al₁₃⁺ cluster, y-component of polarizability (α_yy ) is largest (706.30 unit) and z-component of polarizabilty (α_zz ) is smallest (379.80 unit).There also exist little amount of cross/off-diagonal components of polarizability due to quasi-planarity and eventually principal component of polarizability reduces simultaneously. Isotropic and anisotropic polarizabilities are respectively 591.47 and 345.68 units. These interesting result also nicely explains the plasmonic responds of the clusters. As these clusters has larger vertical dimension and bilateral symmetry axis passes through y-axis, polarizabilities of all the clusters have been found along y-axis. Due to transition from planar structure to quasi-planar structure its off-diagonal or cross component of polarizability (α_xy or α_yz or α_zx )is enhanced and principal component (α_xx or α_yy or α_zz ) of polarizability is discreased. It is found due to charge separation of the cluster from x-y plane.

Table 5

Polarizabilities for pure planar and quasi-planar Al₁₃⁺ clusters.
Clusters	Pure/ Doping	α_xx	α_xy	α_yy	α_xz	α_yz	α_zz	α_iso	α_aniso
Planar	Pure Al₁₃⁺ cluster	812.64	-0.01	837.28	0.00	0.00	279.04	642.99	546.34
	Li doping	798.11	-0.02	864.39	0.00	0.00	271.61	644.70	378.14
	Na doping	807.05	-0.01	907.09	0.00	0.00	277.41	663.85	828.87
	K doping	814.58	-0.02	929.39	0.00	0.00	283.14	675.70	597.18
Quasi-planar	Pure Al₁₃⁺ cluster	804.96	-13.10	788.16	-82.47	45.08	332.14	641.75	492.86
	Li doping	664.21	9.32	677.51	-131.04	69.60	364.02	568.58	400.74
	Na doping	675.53	13.33	693.96	-129.70	68.80	370.51	580.00	256.29
	K doping	688.32	12.98	706.30	-130.47	69.21	379.80	591.47	345.68

Density Of States:

Plot of density of state (Fig. 7) is also the evidence of permanent electric dipole moment (which has also been found from Tables 2 & 3) having larger Transition dipole moment (TDM) [32]. These plots for all these structures show no mirror symmetry between valance band and conduction band - the states within valance band and conduction band are not symmetrical. Also, none of the dotted line passes through 0 eV energy. Interestingly, these dotted line where it intersects on x-axis i.e. energy axis provides the energy corresponds to the excitation energy with highest oscillator strength (Tables 2 & 3). For planar Al₁₃⁺ cluster it is 5.53 eV, for Li doped cluster it is 5.09 eV, for Na doped cluster it is 5.10 eV, for K doped cluster it is 4.99 eV. Similarly, for quasi-planar Al₁₃⁺ cluster it is 5.37 eV, for Li doped cluster it is 5.61 eV, for Na doped cluster it is 5.63 eV, for K doped cluster it is 5.60 eV. For transition state Al₁₃⁺ cluster it is 5.42 eV (Fig. S4).

TDDFT study reveals that in case of small Aluminum linear chains the plamonic character systematically increases with the number of aluminum atoms constituting the cluster. For the cyclic clusters (Al₃ and Al₄) the plamonicity is lower compared to corresponding linear clusters. Among the three Al₁₃⁺ isomers, plasmonicity character are in the order, planar Al₁₃⁺ cluster(CI) > Transition state Al₁₃⁺ cluster(CT) > True minima Al₁₃⁺ cluster(CII). Planar clusters exhibit greater oscillator strength also plasmonic effect than its corresponding quasi-planar clusters (except K doped cluster). In case of planar clusters, mode of transition dipole moment is confined along the symmetry axis (y-axis), but for quasi-planar cluster z-component of TDM roles into play. For this reason additional TDM along z-axis is found. It effectively reduces net transition dipole moment of the clusters along the symmetry axis. TDM is proportional to Plasmonicity of the cluster. In other word, Collectiveness of oscillation is the characteristic of plasmonic nature of the cluster. Due to an additional mode of oscillation of electron cloud, collectiveness of oscillation reduces in case of quasi-planar cluster than planar cluster where z-component of oscillation is totally absent. Next, doping reduces the oscillator strength of the clusters in case of planar clusters. In this case oscillator strengths are in this order: Pure Al₁₃⁺ > K doping > Li doping > Na doping. This can be correlated with the modulation of the dimensions (length and width) of the parent cluster on alkali doping. Alkali atoms having smaller atomic diameter which are closer to adjacent aluminum atoms reduce the oscillator strength and consequently the plasmonicity of the cluster. But, in case of quasi-planar clusters it decreases in the order, pure Al₁₃⁺ < Li doping < Na doping < K doping. In this case alkali atoms are located at top of the centroid of the triangular core of the clusters which is away from mean plane containing aluminum atoms. Also, doping at quasi-planar cluster reduces the dimension of the clusters. These effects may be the reason behind a little enhancement of oscillator strength of the clusters. So, it is clear that after Aluminum atom, K exhibits greater plasmonic effect and Na atom has less plasmonic behavior. Now, the turns for the study of uv-vis spectra. In case of pure and doped Al₁₃⁺ clusters, plasmonic resonance bands are fall in uv-vis region whereas for Ag, Au nanoparticle it falls in infra red region. For planar clusters resonance absorption frequencies are in this order: Pure Al₁₃⁺ > Na doping > Li doping > K doping. For quasi-planar clusters resonance absorption frequencies are in this order: Pure Al₁₃⁺ < K doping < Li doping < Na doping. So, cluster having greater oscillator strength Plasmon absorption band shifts towards high frequency region. From analysis of canonical molecular orbital it is clear that in most cases, plasmonic transitions with higher oscillator strength take place from delocalized \(\pi or \sigma\) orbitals to the orbitals with less delocalization. Polarizability is also a parameter of plamonicity which is directed along the symmetry axis of the cluster (y-axis) similar as TDM. Isotropic polarizability is also proportional to the y component of polarizability. For planar cluster this parameters are in this order : Pure Al₁₃⁺ < Li doping < Na doping < K doping. For quasi-planar cluster it is: Pure Al₁₃⁺ < Li doping < K doping < Na doping respectively. For quasi-planar cluster, isotropic polarizabilities are in this order: Pure Al₁₃⁺> K doping > Na doping > Li doping and anisotropic polarizabilities are in this order: Pure Al₁₃⁺ > K doping > Na doping > Li doping. So, it is clear to say that isotropic polarizability is proportional to plasmonicity. The cluster having greater polarizability indicates a large plasmonic character. On the other hand anisotropic polarizability indicates less plasmonic behavior. Finally, behavior of plasmonicity can be predicted from the spectra of density of states of the clusters. In these cases, the spectrums are not symmetrical along ground state energy. It indicates that there may be existed high value of transition dipole moment (TDM) also in absence of external electric field. Corresponding energy of the barrier of conduction and valance band is exactly matches with transition energy difference with highest oscillator strength.

In summary, oscillator strength, transition dipolemoment and polarizabiltiy can be effectively utilized to characterize the palsmonic character of small mealt nano clusters in a similar manner as that for small organic molecules in the field of molecular plasmonics. Small aluminum clusters which are attractive due to their plasmonic emission in uv-visible region have been characterized with the help of above mentioned molecular plasmonicity parametes. The present study reveals that linear chain of aluminum atoms are most efficient plasmonic emitters compared to cyclic clusters of same nuclearity. The study of plasmonicity in case of Al₁₃⁺ system reveals that planar cluster is having stronger plamonic emission character. Doping of alkali atoms reduces this character. In summary, plasmonic character of small aluminum clusters depend on shape and size of the clusters and the plasmonic emission frequency which fall in the range 170 to 416 nm can be tuned accordingly. The information revealed in this study should become useful in the design of aluminum based plasmonic nano devices.

Funding:

A. D. J. acknowledges the financial support from the Department of Science, Technology and Biotechnology, Government of West Bengal, India through the project memo no: 250(Sanc.)/ST/P/S&T/16G-47/2017 Date: 25/03/2018. S. Guin acknowledges financial support from the UGC India for JRF fellowship ref no 1432/(CSIR-UGC NET DEC-2018).

Conflict of interest/Competing interests:

There is no conflict of interest in the current work.

Availability of data and material: N/A

Code availability: N/A

Authors’ contributions: Surajit Guin has carried out the computational works and prepared the draft manuscript. Atish Dipankar Jana has conceptualized the problem, supervised the work and finalized the manuscript.

Valsecchi C and Brolo A G (2013) Periodic Metallic Nanostructures as Plasmonic Chemical Sensors Langmuir 29:5638. https://doi.org/10.1021/la400085r
M H Grace, Pang Lin, Mullen E H and Fainman Y (2008) Plasmonic Sensing of Biological Analytes Through Nanoholes IEEE Sens. J. 8:2074. DOI: 10.1109/JSEN.2008.2007663
Wang J Y, Tsai F J, Huang J J, Chen C Y, Li N, Kiang Y W and Yang C C (2010) Enhancing InGaN-Based Solar Cell Efficiency through Localized Surface Plasmon Interaction by Embedding Ag Nanoparticles in the Absorbing Layer, Opt. Express 18: 2682. https://doi.org/10.1364/OE.18.002682
Zhang X,Wang P, Zhang Z, Fang Y and Sun M (2014) Plasmon-Driven Sequential Chemical Reactions in an Aqueous Environment, Sci. Rep. 4. https://doi.org/10.1038/srep05407
Gao S, Zhang Z, Liu K and Dong B (2016) Direct Evidence of Plasmonic Enhancement on Catalytic Reduction of 4-Nitrophenol over Silver Nanoparticles Supported on Flexible Fibrous Networks, Appl. Catal. B Environ. 188:245. https://doi.org/10.1016/j.apcatb.2016.01.074
Guenneau S, Ramakrishna S A, Enoch S, Chakrabarti S, Tayeb G and Gralak B (2007) Cloaking and Imaging Effects in Plasmonic Checkerboards of Negative and μ and Dielectric Photonic Crystal Checkerboards Photonics Nanostructures - Fundam. Appl. 5: 63. https://doi.org/10.1016/j.photonics.2007.07.015
Lee Y J, Schade N B, Sun L, Fan J A, Bae D R, Mariscal M M, Lee G, Capasso F, Sacanna S, Manoharan V N and Yi G R (2013) Ultrasmooth, Highly Spherical Monocrystalline Gold Particles for Precision Plasmonics ACS Nano 7:11064. https://doi.org/10.1021/nn404765w
Loiseau A, Asila V, Boitel-Aullen G, Lam M, Salmain M and Boujday S Silver-Based Plasmonic Nanoparticles for and Their Use in Biosensing Biosensors 9. https://doi.org/10.3390/bios9020078
Knight M W, King N S, Liu L, Everitt H O, Nordlander P and Halas N J (2014) Aluminum for Plasmonics ACS Nano 8: 834. https://doi.org/10.1021/nn405495q
Chan G H, Zhao J, Hicks E M, Schatz G C and Van Duyne R P 2007 Plasmonic Properties of Copper Nanoparticles Fabricated by Nanosphere Lithography, Nano Lett. 7 1947.
Agostino S D’and Sala F D, (2013) Handbook of Molecular Plasmonics. https://doi.org/10.1021/nl070648a
Yang K Y, Choi K C and Ahn C W (2009) Surface Plasmon-Enhanced Energy Transfer in an Organic Light-Emitting Device Structure, Opt. Express 17: 11495. https://doi.org/10.1364/OE.17.011495
M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim (2013), Quantum Plasmonics, Nat. Phys. 9:329. https://doi.org/10.1038/nphys2615
Salamon Z, Brown M F and Tollin G 1999 Plasmon Resonance Spectroscopy: Probing Molecular Interactions within Membranes, Trends Biochem. Sci. 24:213. https://doi.org/10.1016/S0968-0004(99)01394-8
Botta R, Upender G, Sathyavathi R, Narayana R D and Bansal C 2013 Silver Nanoclusters Films for Single Molecule Detection Using Surface Enhanced Raman Scattering (SERS), Mater. Chem. Phys. 137:699. https://doi.org/10.1016/j.matchemphys.2012.10.022
Casanova D, Matxain J M and Ugalde J M 2016 Plasmonic Resonances in the Al13- Cluster: Quantification and Origin of Exciton Collectivity, J. Phys. Chem. C 120:12742 . https://doi.org/10.1021/acs.jpcc.6b03210
Kroy K (2014) Levitating Nanoparticles: Non-Equilibrium Nano-Thermometry, Nat. Nanotechnol. 9:415.
Mokkath J H (2017) Shapes Matter: Examining the Optical Response Evolution in Stretched Aluminum Nanoparticles via Time-Dependent Density Functional Theory Phys. Chem. Chem. Phys. 20:51. DOI: 10.1039/C7CP07151E
Guin S, Ghosh SR, Jana AD (2018) First report of a planar and a quasi-planar Al₁₃⁺ cluster having localized antiaromatic deltas within an aromatic sea: NICS, ELF, AIM, and AdNDP bonding analysis. J Mol Model 24:1–14. https://doi.org/10.1007/s00894-018-3875-5
Guin S, Ghosh SR, Jana AD (2020) Planarity does not always mean higher aromaticity – intriguing metalloaromaticity of three Al₁₃⁺ isomers. J Mol Graph Model 97:107544. https://doi.org/10.1016/j.jmgm.2020.107544
Aihara J I and Kanno H (2005) Local Aromaticities in Large Polyacene Molecules, J. Phys. Chem. A 109:3717. https://doi.org/10.1021/jp047183m
Paul M and Balanarayan P (2018) Electron Density and Electrostatic Potential-Based Characteristics of Molecular Plasmons in Polyacenes 19:11.DOI:10.26434/chemrxiv.5539816.v1
M. A. L. Marques and E .K. U. Gross (2004) Time-Dependent Density functional theory Phys. Annual Review of Physical Chemistry 97:0610-00
Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Montgomery JJA, Vreven T, Kudin KN, Burant JC, Millam JM, Iyengar SS, Tomasi J, Barone V, Mennucci B, Cossi M, Scalmani G, Rega N, Petersson GA, Nakatsuji H, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, KleneM LX, Knox JE, Hratchian HP, Cross JB, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Ayala PY, Morokuma K, Voth GA, Salvador P, Dannenberg JJ, Zakrzewski VG, Dapprich S, Daniels AD, Strain MC, Farkas O, Malick DK, Rabuck AD, Raghavachari K, Foresman JB, Ortiz JV, Cui Q, Baboul AG, Clifford S. Cioslowski J, Stefanov BB, Liu G, Liashenko A, Piskorz P, Komaromi I, Martin RL, Fox DJ, Keith T, Al-Laham MA, Peng CY, Nanayakkara A, Challacomb M, Gill PMW, Johnson B, Chen W, Wong MW, Gonzalez C, Pople JA 2004 Gaussian 03 revision C.02. Gaussian Inc Wallingford CT
FELLER D (1995) The Role of Databases in Support of Computational Chemistry Calculations Journal of Computational Chemistry 17:13, 1571 -1586. https://doi.org/10.1002/(SICI)1096-987X(199610)17:13<1571::AID-JCC9>3.0.CO;2-P
Becke A D (1993) A New Mixing of Hartree-Fock and Local Density-Functional Theories, J. Chem. Phys. 98:1372. https://doi.org/10.1063/1.464304
[27] Heard C J, L J Roy (2014) A theoretical study of the structures and optical spectra of helical copper–silver clusters, PCCP. 16:21039. https://doi.org/10.1039/C3CP55507K
Guin S, Jana AD (2021) Effect of alkali atom doping on the electronic structure and aromatic character of planar and quasi-planar Al₁₃⁺ clusters, J. Mol. Model. 27:235. https://doi.org/10.1007/s00894-021-04845-7
[29] Rance G A, Marsh D H, Nicholas R J and Khlobystov A N (2010) UV-Vis Absorption Spectroscopy of Carbon Nanotubes: Relationship between the π-Electron Plasmon and Nanotube Diameter, Chem. Phys. Lett. 493:19. https://doi.org/10.1016/j.cplett.2010.05.012
Corminboeuf C, Heine T and Weber J (2013) Evaluation of Aromaticity: A New Dissected NICS Model Based on Canonical Orbitals, Phys. Chem. Chem. Phys. 5:246. https://doi.org/10.1039/B209674A
Donati V, Lingerfelt D B, Aikens C M and Li X (2018) Anisotropic Polarizability-Induced Plasmon Transfer, J. Phys. Chem. C 122:10621. https://doi.org/10.1021/acs.jpcc.8b02425
Carminati R, Cazé A, Cao D, Peragut F, Krachmalnicoff V, Pierrat R and De Wilde Y (2015) Electromagnetic Density of States in Complex Plasmonic Systems, Surf. Sci. Rep. 70:1. https://doi.org/10.1016/j.surfrep.2014.11.001

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Quantifying plasmonic characteristics of a series of pure Aluminum and alkali doped Aluminum clusters

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Abstract

Figures

Introduction

Tddft And Plamonicity:

Result

A brief introduction of the studied system:

Optimized geometry of small Al clusters

Time Dependent Density Functional Theory (Tddft):

TDDFT study for small Al clusters

Tddft Study For Undoped And Doped Al Clusters

Uv-vis Spectra Study:

UV-Vis spectra for small Al clusters

Canonical Molecular Orbitals:

Canonical Molecular Orbital for small Al clusters

Canonical Molecular Orbital For Small Al Clusters

Polarizability Study:

Density Of States:

Discussions

Conclusions

Declarations

References

Additional Declarations

Supplementary Files

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