Large-scale first principles configuration interaction calculations of optical absorption in aluminum clusters
aa r X i v : . [ phy s i c s . a t m - c l u s ] O c t Large-scale first principles configuration interactioncalculations of optical absorption in aluminum clusters
Ravindra Shinde a ∗ and Alok Shukla b ∗ Department of Physics, Indian Institute of Technology Bombay, Mumbai, Maharashtra 400076,INDIA.
E-mail: [email protected]; [email protected]
Phone: +91 (0)22 25764558. Fax: +91 (0)22 25767552
Abstract
We report the linear optical absorption spectra of aluminum clusters Al n (n=2–5) involvingvalence transitions, computed using the large-scale all-electron configuration interaction (CI)methodology. Several low-lying isomers of each cluster were considered, and their geome-tries were optimized at the coupled-cluster singles doubles (CCSD) level of theory. With theseoptimized ground-state geometries, excited states of different clusters were computed usingthe multi-reference singles-doubles configuration-interaction (MRSDCI) approach, which in-cludes electron correlation effects at a sophisticated level. These CI wave functions were usedto compute the transition dipole matrix elements connecting the ground and various excitedstates of different clusters, and thus their photoabsorption spectra. The convergence of our ∗ To whom correspondence should be addressed † Electronic Supplementary Information (ESI) available: A detailed information about wave functions of excitedstates contributing to various photoabsorption peaks is presented in the supplementary information (Table I throughIX). See DOI: 10.1039/b000000x/ a Department of Physics, Indian Institute of Technology Bombay, Mumbai, Maharashtra 400076, India. Fax: +9122 2576 7552; Tel: +91 22 2576 4558; E-mail: [email protected] b Department of Physics, Indian Institute of Technology Bombay, Mumbai, Maharashtra 400076, India. Fax: +9122 2576 7552; Tel: +91 22 2576 7576; E-mail: [email protected] esults with respect to the basis sets, and the size of the CI expansion, was carefully exam-ined. Our results were found to be significantly different as compared to those obtained usingtime-dependent density functional theory (TDDFT) [Deshpande et al. Phys. Rev. B , 2003, , 035428]. When compared to available experimental data for the isomers of Al and Al ,our results are in very good agreement as far as important peak positions are concerned. Thecontribution of configurations to many body wavefunction of various excited states suggeststhat in most cases optical excitations involved are collective, and plasmonic in nature.
1. Introduction
Metal clusters are promising candidates in the era of nanotechnology. The reason behind growinginterest in clusters lies in their interesting properties and a vast variety of potential technologicalapplications.
Moreover, simple theoretical models can be exploited to describe their properties.Various jellium models have successfully described electronic structures of alkali metal clus-ters, because alkali metals have free valence electrons. This beautifully explains the higher abun-dance of certain clusters. However, in case of aluminum clusters, the experimental results oftenprovide conflicting evidence about the size at which the jellium model would work.
The theo-retical explanation also depends on the valency of aluminum atoms considered. Since s–p orbitalenergy separation in aluminum atom is 4.99 eV, and it decreases with the cluster size, the valencyshould be changed from one to three. Perturbed jellium model, which takes orbital anisotropyinto account, has successfully explained the mass abundance of aluminum clusters.
Shell structure and s–p hybridization in anionic aluminum clusters were probed using photo-electron spectroscopy by Ganteför and Eberhardt, and Li et al . Evolution of electronic structureand other properties of aluminum clusters has been studied in many reports.
Structuralproperties of aluminum clusters were studied using density functional theory by Rao and Jena. An all electron and model core potential study of various Al clusters was carried out by Mar-tinez et al . Upton performed chemisorption calculations on aluminum clusters and reported thatAl is the smallest cluster that will absorb H . DFT alongwith molecular dynamics were used2o study electronic and structural properties of aluminum clusters. Among more recent works,Drebov and Ahlrichs presented a very detailed and systematic study of geometrical structure andelectronic properties of large Al clusters ranging from Al to Al , and their anions and cations.Alipour and Mohajeri performed a comprehensive study of the electronic structure, ionizationpotential, and static and dynamic polarizabilities (at a fixed frequency) of clusters ranging fromAl to Al .Although the photoabsorption in alkali metal clusters has been studied by many authors at var-ious levels of theory, however, very few theoretical calculations of the photoabsorption spectrain aluminum clusters exist. As far experimental studies of optical absorption in aluminumclusters are concerned, several studies have been performed on Al and Al . Never-theless, to the best of our knowledge, no experimental measurements of optical properties of largeraluminum clusters have been performed.Conventional mass spectrometry only distinguishes clusters according to the masses. Hence,theoretical results can be coupled with the experimental measurements of optical absorption, todistinguish between different isomers of a cluster. This is important for clusters of increasinglarger sizes for which several possible isomers exist. We have recently reported results of such cal-culation on small boron clusters. In this paper, we present results of systematic calculations oflinear optical absorption involving transitions among valence states in various low-lying isomers ofsmall aluminum clusters using ab initio large-scale multi-reference singles doubles configurationinteraction (MRSDCI) method. In our group, in the past we have successfully employed the MRS-DCI approach to compute the photoabsorption spectra of a number of conjugated polymers, and boron clusters.
Therefore, it is our intention in this work to test this approach on clustersmade up of larger atoms, namely aluminum, and critically analyze its performance. Furthermore,the nature of optical excitations involved in absorption has also been investigated by analyzing thewave functions of the excited states.Upon comparing calculated optical absorption spectra of Al and Al , we find very good agree-ment with the available experimental data on important peaks. This suggests that the MRSDCI3pproach is equally effective for Al clusters, as it was, say, for boron clusters. For larger clus-ters, for which no experimental data is available, we compare our results with the time-dependentdensity functional theory (TDDFT) based calculations of Deshpande et al. corresponding to theminimum energy configurations, and find significant differences.Remainder of the paper is organized as follows. Next section discusses theoretical and com-putational details of the calculations, followed by section 3, in which results are presented anddiscussed. Conclusions and future directions are presented in section 4. A detailed informationabout the nature of optical excitation, molecular orbitals of clusters, wave functions of excitedstates contributing to various photoabsorption peaks is presented in the supplementary informa-tion.
2. Theoretical and Computational Details
The geometry of various isomers were optimized using the size-consistent coupled-cluster singles-doubles (CCSD) method, as implemented in the
GAUSSIAN
09 package. A basis set of 6-311++G(2d,2p) was used which was included in the
GAUSSIAN
09 package itself. This basisset is optimized for the ground state calculations.We repeated the optimization for singlet and triplet systems on even numbered electron sys-tems to look for the true ground state geometry. Similarly, for odd numbered electron systems,doublet and quartet multiplicities were considered in the geometry optimization. To initiate theoptimization, raw geometries, reported by Rao and Jena, based on density functional method wereused. Figure 1 shows the final optimized geometries of the isomers studied in this paper.Using these optimized geometries, correlated calculations were performed using multirefer-ence singles doubles configuration interaction (MRSDCI) method for both ground state and excitedstates. This method considers a large number of singly- and doubly- substituted configurationsfrom a large number of reference configurations, and, is well suited for both ground and excitedstates calculations. It takes into account the electron correlations which are inadequately repre-sented in single reference ab initio methods. These ground- and excited-state wavefunctions are4urther used to calculate the transition dipole moment matrix elements, which in turn, are utilizedto compute linear optical absorption spectrum assuming a Lorentzian line shape.Various wave functions of the excited states contributing to the peaks in the spectrum obtainedusing a low-level CI calculations were analyzed, and even bigger MRSDCI calculations were per-formed by including more references, if needed. The criteria of choosing a reference configurationin the calculation was based upon the magnitude of the corresponding coefficients in the CI wavefunction of the excited states contributing to a peak in the spectrum. This process was repeated un-til the spectrum converges within acceptable tolerance and all the configurations which contributeto various excited states were included. The typical total number of configurations considered inthe calculations of various isomers is given in Table 1. We have extensively used such approachin performing large-scale correlated calculations of linear optical absorption spectra of conjugatedpolymers, and atomic clusters.
The CI method is computationally very expensive, mainly, because the number of determi-nants to be considered increases exponentially with the number of electrons, and the number ofmolecular orbitals. Calculations on bigger clusters are prohibitive under such circumstances, andare very time consuming even for the clusters considered here. Point group symmetries (D h , andits subgroups) were taken into account, thereby making calculations for each symmetry subspaceindependent of each other. The core of the aluminum atom was frozen from excitations, keepingonly three valence electrons active per atom. Also an upper limit on the number of virtual orbitalswas imposed, to restrict very high energy excitations. The effect of these approximations on thecomputed photoabsorption spectra has been studied carefully, and is presented in the next section.
3. Results and Discussion
In this section, first we present a systematic study of the convergence of our results and various ap-proximations used. In the latter part, we discuss the results of our calculations on various clusters.
In this section we discuss the convergence of photoab-sorption calculations with respect to the choice of the basis set, and the size of the active orbital5 a) Al , D ¥ h , P u (b) Al , D ¥ h , S g (c) Al , D h , A ′ (d) Al , C v , A (e) Al , D ¥ h , S u (f) Al , D h , B g (g) Al , D h , B u (h) Al , C v , A (i) Al , C v , A Figure 1.
Geometry optimized structures of aluminum clusters with point group symmetry andthe electronic ground state at the CCSD level. All numbers are in Å unit.space.
In the literature several optimized basis sets are available for specific purposes, such as ground stateoptimization, excited state calculations etc. We have reported a systematic basis set dependenceof photoabsorption of boron cluster. Similarly, here we have checked the dependence of pho-toabsorption spectrum of aluminum dimer on basis sets used, as shown in Fig. 2. The 6-311type Gaussian contracted basis sets are known to be good for ground state calculations. The cor-relation consistent (CC) basis sets, namely, CC-polarized valence double-zeta and CC-polarizedvalence triple zeta (cc-pVTZ) give a good description of excited states of various systems. Thelatter is found to be more sophisticated in describing the high energy excitations, which were alsoconfirmed using results of an independent TDDFT calculation. Therefore, in this work, we haveused the cc-pVTZ basis set for the optical absorption calculations.6 I n t e n s it y ( a r b . un it s ) CC-PVTZCC-PVDZ6-311++G**6-311++G(2d,2p)6-311++G(3df,3pd)
Figure 2.
Optical absorption in Al calculated using various Gaussian contracted basis sets. I n t e n s it y ( a r b . un it s ) Orbitals upto -4 Ha frozenOrbitals upto -1 Ha frozen
Figure 3.
The effect of freezing the core orbitals of aluminum atoms on optical absorption spec-trum of Al . It renders little effect on optical absorption spectrum, with significant reduction in thecomputational cost. With respect to the total number of orbitals N in the system, the computational time in configura-tion interaction calculations scales as ≈ N . Therefore, such calculations become intractable formoderately sized systems, such as those considered here. So, in order to make those calculationspossible, the lowest lying molecular orbitals are constrained to be doubly occupied in all the con-figurations, implying that no virtual excitation can occur from those orbitals. It reduces the size ofthe CI Hamiltonian matrix drastically. In fact, this approach is recommended in quantum chemicalcalculations, because the basis sets used are not optimized to incorporate the correlations in core7lectrons. The effect of this approximation on the spectrum is as shown in Fig. 3. Since, calcu-lations with all electrons in active orbitals were unfeasible, we have frozen occupied orbitals upto-4 Hartree of energy for the purpose of demonstration. The effect of freezing the core is negligiblysmall in the low energy regime, but shows disagreement in the higher energy range. However,for very high energy excitations, photodissociation may occur, hence absorption spectra at thoseenergies will cease to have meaning. Thus, the advantage of freezing the core subdues this issue.Therefore, in all the calculations presented here, we have frozen the chemical core. I n t e n s it y ( a r b . un it s ) All virtual orbitals activeCutoff 3.0 HaCutoff 1.5 HaCutoff 1.0 HaCutoff 0.8 Ha
Figure 4.
The effect of the number of active orbitals (N act ) on the optical absorption spectrum ofAl . Until N act =46, the optical spectrum does not exhibit any significant change. It corresponds to1.0 Hartree ( ≈ . .1.3 Size of the CI expansionTable 1. The average number of total configurations (N total ) involved in MRSDCI calculations,ground state (GS) energies (in Hartree) at the MRSDCI level, relative energies and correlationenergies (in eV) of various isomers of aluminum clusters.
Cluster Isomer N total
GS energy Relative Correlation energy c (Ha) energy (eV) per atom(eV)Al Linear-I 445716 -483.9138882 0.00 1.69Linear-II 326696 -483.9115660 0.06 1.87Al Equilateral triangular 1917948 -725.9053663 0.00 2.38Isosceles triangular 1786700 -725.8748996 0.83 2.36Linear 1627016 -725.8370397 1.85 2.16Al Rhombus 3460368 -967.8665897 0.00 1.82Square 1940116 -967.8258673 1.11 1.80Al Pentagonal 3569914 -1209.8114803 0.00 1.73Pyramidal 3825182 -1209.7836568 0.76 1.77
In the multi-reference CI method, the size of the Hamiltonian matrix increases exponentiallywith the number of molecular orbitals in the system. Also, accurate correlated results can onlybe obtained if sufficient number of reference configurations are included in the calculations. Inour calculations, we have included those configurations which are dominant in the wave functionsof excited states for a given absorption peak. Also, for ground state calculations, we includedconfigurations until the total energy converges within a predefined tolerance. Table 1 shows theaverage number of total configurations involved in the CI calculations of various isomers. For agiven isomer, the average is calculated across different irreducible representations needed in thesesymmetry adapted calculations of the ground and various excited states. For the simplest cluster,the total configurations are about half a million and for the biggest cluster considered here, it isaround four million for each symmetry subspace of Al . The superiority of our calculations canalso be judged from the correlation energy defined here ( cf. Table 1), which is the difference in thetotal energy of a system at the MRSDCI level and the Hartree-Fock level. The correlation energy c The difference in Hartree-Fock energy and MRSDCI correlated energy of the ground state.
In this section, we describethe photoabsorption spectra of various isomers of the aluminum clusters studied. Plots of variousmolecular orbitals involved are presented in the Electronic Supporting Information (ESI).† Aluminum dimer is the most widely studied cluster of aluminum, perhaps because the nature ofits ground state was a matter of debate for a long time. For example, in an early emission basedexperiment Ginter et al. concluded that ground state of Al was of symmetry S − u , while in alow-temperature absorption based experiment Douglas et al. deduced that the ground state ofthe system was of S − g . In other words, even the spin multiplicity of the cluster was measured to bedifferent in different experiments. Theoreticians, on the other hand, were unanimous in predictingthe spin multiplicity of the ground state to be of triplet type, however, some predicted P u to be theground state, while others predicted it to be of S − g type. Perhaps, the reason behindthis ambiguity, was that states P u and S − g are located extremely close to each other as discoveredin several theoretical calculations. However, it has now been confirmed experimentallyby Cai et al. and Fu et al. that the Al ( cf . Fig. 1(a)) has P u ground state, with the S − g statebeing a metastable state located slightly above it.In our calculations, the bond length obtained using geometry optimization at CCSD level was2.72 Å, with D ¥ h point group symmetry. This is in very good agreement with available data, suchas Martinez et al . obtained 2.73 Å as dimer length using all electron calculations, and2.75 Å as bond lengths using DFT and configuration interaction methods, and 2.86 Å obtainedusing DFT with generalized gradient approximation. The experimental bond length of aluminumdimer is 2.70 Å. We also performed the geometry optimization for the metastable state S − g mentioned above, and found the bond length to be 2.48 Å ( cf . Fig. 1(a)). Using MRCI calculationsBauschlicher et al. estimated that S − g electronic state lies 0.02 eV above the P u ground state. for the P u ground state consists of two degeneratesingly occupied molecular orbitals (to be denoted by H and H , henceforth), because it is a spintriplet system. Similarly, the configurations involving excitations from occupied molecular orbitalsto the unoccupied orbitals, form excited state wave functions. The computed photoabsorptionspectra of Al , as shown in Fig. 5, is characterized by weaker absorption at lower energies andcouple of intense peaks at higher energies. The many-particle wave functions of excited statescontributing to the peaks are presented in Table I of supporting information. The spectrum startswith a small absorption peak (I k ) at around 2 eV, characterized by H → L + k ,III ⊥ ), until a dominant absorption (IV k ) is seen at 5 eV. This is characterized by H → L + ⊥ ) is observed at 8 eV having H − → L as dominant configuration,with absorption due to light polarized perpendicular to the axis of the dimer.The optical absorption spectrum of metastable dimer in the S − g state ( cf. Fig. 5) is alsocharacterized by small absorption peaks in the lower energy range. Also, all peaks of the spectrumappear blue-shifted as compared to that of stable isomer. The peak (I k ) at 2.29 eV is characterizedby H − → L , while two major peaks at 5.17 eV (V k ) and 8.13 eV (X ⊥ ) are characterized by H − → L configuration due to light polarized along the direction of axis of dimer and H − → L + et al. obtained the low-energy optical absorption in the cryogenic krypton matrix.The major peaks in this experimental absorption spectrum at 1.77 eV and 3.13 eV can be associatedwith our results of 1.96 eV and 3.17 eV respectively. Although, our calculation overestimates thelocation of the first peak by about 11%, the agreement between theory and experiment is excellentfor the second peak, giving us confidence about the quality our calculations. Furthermore, thecomputed spectrum for the metastable state S − g of Al ( cf. Fig. 5) has no peaks close to thoseobserved in the experiments, implies that measured optical absorption occurs in the P u state ofthe system, confirming that the ground state has P u symmetry.11ur spectrum differs from the one obtained with the time-dependent local density approxima-tion (TDLDA) method in both the intensity and the number of peaks. However, we agree withTDLDA in predicting two major peaks at 5 eV (IV k ) and 8 eV (VIII ⊥ ). Unlike our calculations,the number of peaks is much more in TDLDA results and the spectrum is almost continuous. Peakslocated in our calculations at 3.2 eV (II k ) and 6.3 eV (V ⊥ ) are also observed in the TDLDA spec-trum of the dimer, except for the fact that in our calculations both the peaks are relatively minor,while the TDLDA calculation predicts the 6.3 eV peak to be fairly intense. Figure 5.
The linear optical absorption spectra of the global minimum Al isomer ( P u state, toppanel) and metastable isomer ( S − g state, bottom panel), calculated using the MRSDCI approach.The peaks corresponding to the light polarized along the molecular axis are labeled with the sub-script k , while those polarized perpendicular to it are denoted by the subscript ⊥ . For plotting thespectrum, a uniform linewidth of 0.1 eV was used.12 .2.2 Al Among the possible isomers of aluminum cluster Al , the equilateral triangular isomer is foundto be the most stable. We have considered three isomers of Al , namely, equilateral triangle,isosceles triangle, and a linear chain. The most stable isomer has D h point group symmetry,and A ′ electronic state. The optimized bond length 2.57 Å, is in good agreement with reportedtheoretical values 2.61 Å, and 2.52 Å. The doublet ground stateis also confirmed with the results of magnetic deflection experiments. The next isomer, which lies 0.83 eV higher in energy, is the isosceles triangular isomer. Theoptimized geometry has 2.59 Å, 2.59 Å and 2.99 Å as sides of triangle, with a quartet ground state( A ). Our results are in agreement with other theoretical results. Linear Al isomer again with quartet multiplicity is the next low-lying isomer. The optimizedbond length is 2.62 Å. This is in good agreement with few available reports. Li et al. reported infrared optical absoption in Al in inert-gas matrices at low temperature. Another experimental study of optical absorption in isosceles triangular isomer was performed byFu et al. using jet cooled aluminum clusters.
The photoabsorption spectra of these isomers are presented in Fig. 6. The corresponding manybody wave functions of excited states corresponding to various peaks are presented in Table III,IV and V of supporting information. In the equilateral triangular isomer, most of the intensity isconcentrated at higher energies. The same is true for the isosceles triangular isomer. However, thespectrum of isosceles triangular isomer appears slightly red shifted with respect to the equilateralcounterpart. Along with this shift, there appears a split pair of peaks at 5.8 eV (VI and VII).This splitting of oscillator strengths is due to distortion accompanied by symmetry breaking. Theabsorption spectrum of linear isomer is altogether different with bulk of the oscillator strengthcarried by peaks in the range 4 – 5 eV, and, due to the polarization of light absorbed parallel to theaxis of the trimer. 13 igure 6.
The linear optical absorption spectra of Al equilateral triangle isomer, isosceles iso-mer, and linear isomer calculated using the MRSDCI approach. The peaks corresponding to thelight polarized along the molecular plane are labeled with the subscript k , while those polarizedperpendicular to it are denoted by the subscript ⊥ . All peaks in the spectrum of isosceles isomercorrespond to the light polarized along the molecular plane. Rest of the information is same asgiven in the caption of Fig. 5 14he optical absorption spectrum of equilateral triangular isomer consists of very feeble lowenergy peaks at 3.5 eV (I k ), 5.6 eV (II k ) and 5.8 eV (III ⊥ ) characterized by H − → L +
5, adouble excitation H − → L + H − → L +
5, and H − → L + k ) at around 6.5 eV with dominant contribution from H → L + H → L + k ) is observed at 7.5 eV characterized mainly due to doubleexcitations.Two major peaks at 6.5 eV (IV k ) and 7.5 eV (VI k ) in the spectrum of Al equilateral isomer,obtained in our calculations are also found in the spectrum of TDLDA calculations, with the dif-ference that the latter does not have a smaller intensity in TDLDA. Other major peaks obtainedby Deshpande et al. in the spectrum of aluminum trimer are not observed, or have very smallintensity in our results.As compared to the equilateral triangle spectra, the isosceles triangular isomer with quartetspin multiplicity, exhibits several small intensity peaks ( cf . Fig. 6) in the low energy regime. Themajority of contribution to peaks of this spectrum comes from in-plane polarized transitions, withnegligible contribution from transverse polarized light. The spectrum starts with a feeble peak (I k )at 2.4 eV with contribution from doubly-excited configuration H → L + H − → L +
2. Al-though, no experimental absorption data is available for the doublet equilateral triangle isomer, Fu et al. managed to measure the absorption of the isosceles triangle isomer, and observed thispeak to be around 2.5 eV. Thus, this excellent agreement between the experiment and our theo-retical calculations for isosceles triangle isomer with quartet spin multiplicity, further strengthensour belief in the quality of our calculations. One of the dominant contribution to the oscillatorstrength comes from two closely-lying peaks (VI k and VII k ) at 5.8 eV. The wave functions ofexcited states corresponding to this peak show a strong mixing of doubly-excited configurations,such as H − → L + H − → L and H − → L + H − → L . The peak (VIII k ) at 6.7 eVshows absorption mainly due to H → L + k ) and 2.3 eV (II k ), both characterized by H − → H −
2. Thisconfiguration also contributes to the semi-major peak (III k ) at 4 eV along with H − → H . Twoclosely lying peaks at 4.3 eV (IV k , ⊥ ) and 4.6 eV (V k ) carry the bulk of the oscillator strength.Major contribution to the former comes from H − → L + H − → H − Figure 7.
The linear optical absorption spectra of rhombus and square isomers of Al , calculatedusing the MRSDCI approach. Rest of the information is same as given in the caption of Fig. 6.Tetramer of aluminum cluster has many low lying isomers due to its flat potential energy curves.Among them, rhombus structure is the most stable with B g electronic ground state. Our opti-16ized bond length for rhombus structure is 2.50 Å and 63 . ◦ as the acute angle. This is to becompared with corresponding reported values of 2.56 Å and 69 . ◦ reported by Martinez et al. , . ◦ computed by Jones, . ◦ obtained by Schultz et al . We note thatbond lengths are in good agreement but bond angles appear to vary a bit.The other isomer studied here is a square shaped tetramer with optimized bond length of 2.69Å. The electronic ground state of this D h symmetric cluster is B u . This optimized geometry isin accord with 2.69 Å reported by Martinez et al ., however, it is somewhat bigger than 2.57 Åcalculated by Yang et al. and 2.61 Å obtained by Jones. For planar clusters, like rhombus and square shaped Al , two types of optical absorptions arepossible: (a) planar – those polarized in the plane of the cluster, and (b) transverse – the ones polar-ized perpendicular to that plane. The many-particle wave functions of excited states contributingto the peaks are presented in Table VI and VII of supporting information. The onset of opticalabsorption in rhombus isomer occurs at around 1 eV (I ⊥ ) with transversely polarized absorptioncharacterized by H → L +
1. It is followed by an in-plane polarized absorption peak (II k ) at 2.3eV with dominant contribution from H − → H . Several closely lying peaks are observed in asmall energy range of 4.5 – 8 eV. Peaks split from each other are seen in this range confirming thatafter shell closure, in perturbed droplet model, Jahn Teller distortion causes symmetry breakingusually associated with split absorption peaks. The most intense peak (V k ) is observed at 5.5 eVcharacterized by H − → L + k ) and 2.7 eV (II k ) characterized by H − → L and H → L + k ) and 4.9 eV (IV k ) have H − → L and H → L + k ) at 5.85 eV is observed with absorption due toin-plane polarization having H − → L + H → L + H − → L + k , ⊥ ) at 6.5 eV. This peak along with one at 6.9 eV (VIII k , ⊥ ) are two equally and most in-tense peaks of the spectrum. The latter has additional contribution from the double excitation17 → L + H − → L . A shoulder peak (IX k ) is observed at 7.2 eV.The TDLDA spectrum of aluminum rhombus tetramer differs from the one presented here.Peaks labeled III to XII in our calculated spectrum are also observed in the TDLDA results, however, the relative intensities tend to disagree. For example, the strongest absorption peak ofTDLDA calculations is located around 7.9 eV, while in our spectrum we obtain the second mostintense peak at that location. The highest absorption peak (V k ) in our calculations is at 5.5 eV,while TDLDA does report a strong peak at the same energy, it is not the highest of the spectrum.Our calculations also reveal a strong structure-property relationship as far as the location of themost intense peak in the absorption spectra of the two isomers is considered, a feature which canbe utilized in their optical detection. The lowest lying pentagonal isomer of aluminum has C v symmetry and has an electronic groundstate of A . The bond lengths are as shown in Fig. 1(h). These are slightly bigger than thoseobtained by Rao and Jena and Yang et al. using the DFT approach. Many other reports haveconfirmed that the planar pentagon is the most stable isomer of Al .The other optimized structure of pentamer is perfect pyramid with C v symmetry and A elec-tronic ground state. This lies 0.76 eV above the global minimum structure. This is the only threedimensional structure studied in this paper for optical absorption. The optimized geometry is con-sistent with those reported earlier by Jones. However, it should be noted that there exists manymore similar or slightly distorted structure lying equally close the the global minimum.The many-particle wave functions of excited states contributing to the peaks are presented inTable VIII and IX of supporting information. The optical absorption spectrum of pentagonal Al (Fig. 8) has few low energy peaks followed by major absorption (V k ) at 4.4 eV. It has dominantcontribution from H − → L + igure 8. The linear optical absorption spectra of pentagonal and pyramidal Al , calculated usingthe MRSDCI approach. The peaks in the spectrum of pyramidal isomer corresponding to the lightpolarized along the Cartesian axes are labeled accordingly. Cartesian xy plane is assumed parallelto the base of the pyramid. Rest of the information is same as given in the caption of Fig. 6.The major absorption peak (V x , y ) at 4.2 eV is slightly red-shifted as compared to the pentagonalcounterpart. It is characterized by H − → L +
2. A peak (X x ) at 6 eV is seen in this absorptionspectrum having dominant contribution from H → L +
13, which is missing in the spectrum ofpentagon. These differences can lead to identification of isomers produced experimentally.In the range of spectrum studied in our calculations, the TDLDA calculated spectrum ofpentagonal isomer is found to be similar to the one presented here as far as the peak locations areconcerned, albeit the intensity profile differs at places. A small peak at 2.4 eV (II k ) is observedin both the spectra, followed by peaks at 3.9 eV (III k , ⊥ ), 4.2 eV (IV k ) and 4.4 eV (V k ). Thesethree peaks are also observed in TDLDA results with a little bit of broadening. Again, the peakat 5.4 eV (VII ⊥ ) matches with each other calculated from both the approaches. Peak found at 6.719V (IX ⊥ ) is also observed in the TDLDA calculation. Within the energy range studied here, thestrongest peak position and intensity of this work is in good agreement with that of its TDLDAcounterpart.
4. Conclusions and Outlook
In this study, we have presented large-scale all-electron correlated calculations of optical absorp-tion spectra of several low-lying isomers of aluminum clusters Al n (n=2–5), involving valencetransitions. The present study does not take into account Rydberg transitions, which are more ofatomic properties, than molecular ones. Both ground and excited state calculations were performedat MRSDCI level, which take electron correlations into account at a sophisticated level. We haveanalyzed the nature of low-lying excited states. We see strong configuration mixing in variousexcited states indicating plasmonic nature of excitations as per the criterion suggested by Blanc etal. Isomers of a given cluster show a distinct signature spectrum, indicating a strong structure-property relationship, which is usually found in small metal clusters. Such structure-propertyrelationship exists for photoelectron spectroscopy as well, therefore, the optical absorption spec-troscopy can be used as an alternative probe of the structures of clusters, and can be employed inexperiments to distinguish between different isomers of a cluster. The optical absorption spectraof few isomers of aluminum dimer and trimer are in very good agreement with the available exper-imental results. Owing to the sophistication of our calculations, our results can be used for bench-marking of the absorption spectra. Furthermore, our calculations demonstrate that the MRSDCIapproach, within a first-principles formalism, can be used to perform sophisticated calculationsof not just the ground state, but also of the excited states of metal clusters, in a numerically effi-cient manner. Moreover, by using more diffuse basis functions, one can also compute the Rydbergtransitions, in case their description is warranted.Our results were found to be significantly different as compared to the TDLDA results, forthe clusters studied here. This disagreement could be resolved by future optical absorption experi-20ents performed on these clusters. Acknowledgments
One of us (R.S.) would like to acknowledge the Council of Scientific and Industrial Research(CSIR), India, for research fellowship (09/087/(0600)2010-EMR-I). We also acknowledge CDAC,Pune for providing computational facility Param Yuva -II. .
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5. Molecular Orbitals of Aluminum Clusters
Figure 9.
Molecular orbitals of aluminum dimer. H and L stands for HOMO and LUMO respec-tively, and H and H are singly occupied degenerate molecular orbitals.25 igure 10. Molecular orbitals of equilateral triangular aluminum trimer. H and L stands forHOMO and LUMO respectively. ( H − H − L , L +
1) and ( L + L +
3) are degenerate pairs.
Figure 11.
Molecular orbitals of isosceles triangular aluminum trimer. H and L stands for HOMOand LUMO respectively, and H , H , and H are singly occupied molecular orbitals. Figure 12.
Molecular orbitals of linear aluminum trimer. H and L stands for HOMO and LUMOrespectively, and H , H , and H are singly occupied molecular orbitals. Figure 13.
Molecular orbitals of rhombus-shaped aluminum tetramer. H and L stands for HOMOand LUMO respectively, and H and H are singly occupied molecular orbitals.26 igure 14. Molecular orbitals of square-shaped aluminum tetramer. H and L stands for HOMOand LUMO respectively, and H and H are singly occupied molecular orbitals. Figure 15.
Molecular orbitals of pentagonal aluminum pentamer. H and L stands for HOMO andLUMO respectively. Figure 16.
Molecular orbitals of pyramidal aluminum pentamer. H and L stands for HOMO andLUMO respectively. 27 able 2. Excitation energies ( E ) and many-particle wave functions of excited states correspondingto the peaks in the linear absorption spectrum of Al , along with the oscillator strength ( f ) ofthe transitions. Longitudinal and transverse polarization corresponds to the absorption due to lightpolarized along and perpendicular to the molecular axis respectively. In the wave function, thebracketed numbers are the CI coefficients of a given electronic configuration. Symbols H , H denote SOMOs discussed earlier, and H , and L , denote HOMO and LUMO orbitals respectively. HF denotes the Hartree-Fock configuration.Peak E (eV) Symmetry f Polarization Wave FunctionGS B u | H , H i (0.9096) | H − → H ; H → L i (0.1139) | H − → L ; H − → L + i (0.0889)I 1.96 B g | H → L + i (0.8120) | H − → H i (0.3685)II 3.17 B g | H − → H i (0.6172) | H → L + i (0.4068) | H → L ; H − → L i (0.3190)III 4.47 A g | H → L + i (0.8313) | H → L + i (0.2024)IV 4.99 B g | H → L + i (0.7353) | H − → H i (0.4104)V 6.31 A g | H → L + i (0.4683) | H − → L + i (0.3894) | H − → L ; H → L + i (0.3886)VI 7.17 A g | H → L + H − → L i (0.4782) | H − → L + i (0.4327) | H → L ; H → L + i (0.3867)VII 7.79 A g | H − → H ; H → L + i (0.4833) | H → L + i (0.3917) | H → L ; H → L + i (0.3791)VIII 8.05 B g | H − → L i (0.5316) | H − → L + i (0.3756) | H → L + i (0.3531)8.10 A g | H − → H ; H → L + i (0.4788) | H → L + i (0.4095)IX 8.87 B g | H → L + i (0.5061) | H → L ; H → L + i (0.4162)8.95 A g | H → L + i (0.4932) | H → L ; H → L + i (0.4414) | H → L + H − → L + i (0.3262)28 able 3. Excitation energies ( E ) and many-particle wave functions of excited states correspondingto the peaks in the linear absorption spectrum of metastable Al , along with the oscillator strength( f ) of the transitions. Longitudinal and transverse polarization corresponds to the absorption dueto light polarized along and perpendicular to the molecular axis respectively. In the wave function,the bracketed numbers are the CI coefficients of a given electronic configuration. Symbols H , H denote SOMOs discussed earlier, and H , and L , denote HOMO and LUMO orbitals respectively. HF denotes the Hartree-Fock configuration.Peak E (eV) Symmetry f Polarization Wave FunctionGS B g | H , H i (0.8975) | H − → L ; H − → L i (0.1418) | H − → L ; H → L + i (0.1146)I 2.29 A u | H − → L i (0.6598) | H → L + i (0.4276) | H → L + i (0.4276)II 3.26 A u | H − → L i (0.7659) | H → L i (0.3137)III 4.40 B u , u | H − → L + i (0.5540) | H → L + H − → H i (0.4827)IV 4.67 B u , u | H → L + i (0.5073) | H − → L + i (0.5030)V 5.17 A u | H − → L i (0.7286) | H → L + i (0.3078)VI 5.75 B u , u | H − → H i (0.5354) | H − → L + i (0.4847)VII 6.24 B u , u | H − → L + i (0.5856) | H − → L ; H − → H i (0.3432)VIII 6.79 B u , u | H → L ; H → L + i (0.4766) | H → L ; H − → L + i (0.4333)IX 7.73 A u | H − → L + i (0.6484) | H → L + H − → L i (0.2333)X 8.13 B u , u | H − → L + i (0.4767) | H → L + i (0.4052)XI 8.49 B u , u | H → L ; H − → L + i (0.4727) | H − → H i (0.3364)29 able 4. Excitation energies ( E ) and many-particle wave functions of excited states correspondingto the peaks in the linear absorption spectrum of Al equilateral triangle isomer, along with theoscillator strength ( f ) of the transitions. In-plane and transverse polarization corresponds to theabsorption due to light polarized in and perpendicular to the plane of the triangular isomer respec-tively. In the wave function, the bracketed numbers are the CI coefficients of a given electronicconfiguration. Symbols H and L , denote HOMO (singly occupied, in this case) and LUMO orbitalsrespectively. HF denotes the Hartree-Fock configuration.Peak E (eV) Symmetry f Polarization Wave FunctionGS A | HF i (0.8373) | H − → L + i (0.1329)I 3.42 B | H − → L + i (0.2908) | H − → L + i (0.2439)3.54 A | H − → L + i (0.3686) | H − → H i (0.3403)II 5.61 A | H − → L + H − → L + i (0.4854) | H → L + H − → L + i (0.4476)III 5.87 B | H − → L + i (0.2915) | H − → L i (0.2842)IV 6.53 A | H → L + i (0.4044) | H − → L + i (0.3965) | H − → L + i (0.3158)6.53 B | H → L + i (0.3842) | H − → L + i (0.2834) | H − → L + i (0.2256)V 6.96 B | H − → L i (0.3140) | H − → L + i (0.2626)VI 7.50 B | H − → L + H → L + i (0.3136) | H − → L + i (0.2864)7.57 A | H → L + H − → L + i (0.3838) | H − → L + i (0.2651) | H − → L + i (0.2590)30 able 5. Excitation energies ( E ) and many-particle wave functions of excited states correspond-ing to the peaks in the linear absorption spectrum of Al isosceles triangle isomer, along with theoscillator strength ( f ) of the transitions. In-plane and transverse polarization corresponds to theabsorption due to light polarized in and perpendicular to the plane of the triangular isomer respec-tively. In the wave function, the bracketed numbers are the CI coefficients of a given electronicconfiguration. Symbols H , H and H denote SOMOs discussed earlier, H and L , denote HOMOand LUMO orbitals respectively.Peak E (eV) Symmetry f Polarization Wave FunctionGS A | H , H , H i (0.8670) | H − → L + i (0.1213)I 2.37 A | H → L + H → L + i (0.7066) | H − → L + H → L i (0.4052)II 3.06 B | H → H ; H − → L i (0.4691) | H − → L + H → H i (0.4070)III 3.45 A | H → L + i (0.5566) | H − → L + H → L i (0.5209)IV 4.11 B | H → L + i (0.6038) | H → L + H − → L i (0.5272)V 4.83 A | H → L + H − → L + i (0.5321) | H → L + i (0.2611)VI 5.76 A | H − → L + H → L i (0.3479) | H − → L + H → L i (0.2875) | H → L + H → L + i (0.2800)5.85 B | H → L + H − → L i (0.4081) | H − → L ; H → L i (0.2400)VII 5.95 A | H − → L + i (0.3296) | H − → L + H → L i (0.3138)6.15 B | H → L + i (0.7827)VIII 6.68 B | H → L + i (0.4548) | H → L + H → L + i (0.2705) | H → L + i (0.2447)31 able 6. Excitation energies ( E ) and many-particle wave functions of excited states correspondingto the peaks in the linear absorption spectrum of Al linear isomer, along with the oscillator strength( f ) of the transitions. Longitudinal and transverse polarization corresponds to the absorption dueto light polarized along and perpendicular to the axis of the linear isomer respectively. In thewave function, the bracketed numbers are the CI coefficients of a given electronic configuration.Symbols H , H and H denote SOMOs discussed earlier, H and L , denote HOMO and LUMOorbitals respectively. HF denotes the Hartree-Fock configuration.Peak E (eV) Symmetry f Polarization Wave FunctionGS A u | H , H , H i (0.8010) | H − → H ; H → L i (0.1913)I 1.24 B g | H → L + i (0.6602) | H − → H i (0.3636)II 2.25 B g | H − → H i (0.6856) | H − → H i (0.3230)III 4.01 B g | H − → H i (0.5249) | H − → H i (0.3471)IV 4.43 B g | H − → H i (0.4070) | H − → L + H → L + i (0.2409)4.47 B g , g | H → L + i (0.5402) | H − → H ; H → L + i (0.3068)V 4.62 B g | H − → H i (0.4600) | H − → L + H → L + i (0.2862)VI 5.29 B g , g | H → L + i (0.4951) | H − → H ; H − → L + i (0.3284) | H − → L + i (0.3091)VII 5.83 B g | H − → L + H → L i (0.6637) | H − → H i (0.2225) | H − → H i (0.2073)VIII 6.31 B g | H → L + H → L i (0.5099) | H → L ; H → L + i (0.2706)6.37 B g , g | H − → L + i (0.3989) | H − → H ; H → L + i (0.2266)IX 6.89 B g | H − → L + i (0.3920) | H → L + H → L + i (0.3086)32 able 7. Excitation energies ( E ) and many-particle wave functions of excited states correspondingto the peaks in the linear absorption spectrum of Al rhombus isomer, along with the oscillatorstrength ( f ) of the transitions. In-plane and transverse polarization corresponds to the absorptiondue to light polarized in and perpendicular to the plane of the rhombus isomer respectively. In thewave function, the bracketed numbers are the CI coefficients of a given electronic configuration.Symbols H , H denote SOMOs discussed earlier, and H , and L , denote HOMO and LUMO orbitalsrespectively.Peak E (eV) Symmetry f Polarization Wave FunctionGS B g | H , H i (0.8724) | H − → L ; H − → L i (0.1050)I 1.07 B u | H → L + i (0.8489) | H − → L + i (0.1601)II 2.31 B u | H − → H i (0.7645) | H → L + i (0.3113)III 4.67 B u | H − → L ; H − → L + i (0.6036) | H − → L + i (0.4213) | H → L + i (0.3113)IV 4.88 A u | H − → L ; H − → L + i (0.6036) | H − → L i (0.4699)V 5.51 B u | H − → L + i (0.7378) | H − → H i (0.2161)VI 5.84 A u | H − → L + i (0.3889) | H − → L ; H − → L i (0.3758) | H − → L i (0.3594) | H − → L ; H − → L + i (0.3591)VII 6.01 B u | H → L + i (0.7268) | H − → L + i (0.3050)VIII 6.20 A u | H − → L + i (0.5195) | H − → L ; H − → L i (0.4189)IX 6.51 B u | H − → L + i (0.7001) | H − → H ; H − → L + i (0.2232) | H − → L ; H − → L + i (0.2070)X 6.92 B u | H − → L + i (0.5144) | H − → L ; H − → L + i (0.3549) | H − → L + i (0.2676)XI 7.31 B u | H − → L + i (0.4033) | H − → L ; H − → L + i (0.3787) able 8. Excitation energies ( E ) and many-particle wave functions of excited states correspondingto the peaks in the linear absorption spectrum of Al square isomer, along with the oscillatorstrength ( f ) of the transitions. In-plane and transverse polarization corresponds to the absorptiondue to light polarized in and perpendicular to the plane of the rhombus isomer respectively. In thewave function, the bracketed numbers are the CI coefficients of a given electronic configuration.Symbols H , H denote SOMOs discussed earlier, and H , and L , denote HOMO and LUMO orbitalsrespectively.Peak E (eV) Symmetry f Polarization Wave FunctionGS B u | H , H i (0.8525) | H → L ; H − → L i (0.0972)I 2.08 B g , g | H − → L i (0.7191) | H − → H ; H → L + i (0.2645) | H → L + i (0.2536) | H → L i (0.2443)II 2.68 B g , g | H → L + i (0.4757) | H − → L i (0.4358) | H − → H ; H → L + i (0.3608)III 4.19 B g , g | H − → L i (0.5889) | H − → L + i (0.4283) | H → L i (0.2329)IV 4.92 B g , g | H → L + i (0.5780) | H − → L + i (0.4083) | H − → L i (0.3198)V 5.17 A g | H − → L ; H → L + i (0.3693) | H − → L ; H → L + i (0.3692)5.33 B g , g | H − → H ; H − → L i (0.5193) | H − → L + i (0.3915) | H − → L + i (0.3335)VI 5.85 B g , g | H − → L + i (0.7184) | H − → H ; H − → L + i (0.2587) | H − → L + i (0.2579)VII 6.55 B g , g | H − → L + i (0.5706) | H − → H ; H − → L + i (0.4089) | H − → L + i (0.3325)6.58 A g | H → L + H − → L i (0.4375) | H → L + H − → L i (0.4375) | H − → L + i (0.4183)VIII 6.87 B g , g | H − → L + i (0.5100) | H − → L + i (0.3495)6.93 A g | H → L + H − → L i (0.3558) | H → L + H − → L i (0.3558)34 able 9. Excitation energies ( E ) and many-particle wave functions of excited states correspondingto the peaks in the linear absorption spectrum of Al pentagonal isomer, along with the oscillatorstrength ( f ) of the transitions. In-plane and transverse polarization corresponds to the absorptiondue to light polarized in and perpendicular to the plane of the pentagonal isomer respectively. In thewave function, the bracketed numbers are the CI coefficients of a given electronic configuration.Symbols H and L , denote HOMO and LUMO orbitals respectively.Peak E (eV) Symmetry f Polarization Wave FunctionGS A | ( H − ) i (0.8679) | H − → L + H → L + i (0.1045)I 1.03 B | H − → L i (0.8635) | H − → L ; H → L + i (0.0880)II 2.38 B | H − → H − i (0.8560) | H − → L + i (0.1387)III 3.90 B | H → L + i (0.8387) | H → L ; H − → L + i (0.1944) A | H − → L i (0.8140) | H − → L + i (0.1841)IV 4.16 B | H − → L + i (0.7276) | H − → L + i (0.4478)V 4.42 B | H − → L + i (0.7096) | H − → L + i (0.4490) | H − → L + i (0.1535)VI 4.78 A | H − → L + i (0.7992) | H − → L ; H → L + i (0.2058)VII 5.46 B | H → L + i (0.8156) | H → L ; H − → L i (0.1708)VIII 6.37 B | H − → L i (0.7632)IX 6.73 B | H − → L i (0.7370) | H → L + i (0.3698) | H − → L ; H → L + i (0.1225)X 7.49 A | H → L + i (0.5087) | H − → L + i (0.3508) | H → L ; H − → L + i (0.2937)35 able 10. Excitation energies ( E ) and many-particle wave functions of excited states correspond-ing to the peaks in the linear absorption spectrum of Al pyramid isomer, along with the oscillatorstrength ( f ) of the transitions. In the wave function, the bracketed numbers are the CI coeffi-cients of a given electronic configuration. Symbols H and L , denote HOMO and LUMO orbitalsrespectively.Peak E (eV) Symmetry f Polarization Wave FunctionGS A | ( H − ) i (0.8591) | H − → L + H − → L + i (0.1138)I 1.72 B | H − → L + i (0.6849) | H − → L + i (0.2887)1.75 A | H → L + i (0.2887)II 2.21 B | H − → L + i (0.7170) | H − → L + i (0.3402) | H − → L + i (0.2290)III 2.55 A | H → L + i (0.5390) | H − → H − i (0.1296)IV 3.46 B | H − → L ; H − → L + i (0.6131) | H − → L + i (0.4975)3.48 A | H − → H − i (0.7340) | H − → L i (0.3735)V 4.04 B | H → L + i (0.5929) | H → L + i (0.4432)4.22 B | H − → L + i (0.8272) | H − → L + i (0.1580)VI 4.74 B | H → L i (0.7617) | H → L + i (0.2542)VII 5.08 A | H − → L ; H → L + i (0.5540) | H − → L i (0.4833)VIII 5.26 A | H − → L ; H → L + i (0.6251) | H − → L i (0.3902)5.27 B | H − → H − H → L + i (0.6056) | H → L i (0.3242)IX 5.56 B | H → L + i (0.7819) | H → L + i (0.3051)X 6.00 B | H → L + i (0.8132) | H → L + ii