First principles electron-correlated calculations of optical absorption in magnesium clusters
FFirst principles electron-correlated calculations of opticalabsorption in magnesium clusters
Ravindra Shinde ∗ Materials Research Center, Indian Institute of Science, Bangalore 560012, India.
Alok Shukla † Department of Physics, Indian Institute ofTechnology Bombay, Mumbai 400076, India. (Dated: August 28, 2018)
Abstract
In this paper, we report large-scale configuration interaction (CI) calculations of linearoptical absorption spectra of various isomers of magnesium clusters Mg n (n=2–5), cor-responding to valence transitions. Geometry optimization of several low-lying isomers ofeach cluster was carried out using coupled-cluster singles doubles (CCSD) approach, andthese geometries were subsequently employed to perform ground and excited state calcu-lations using either the full-CI (FCI) or the multi-reference singles-doubles configurationinteraction (MRSDCI) approach, within the frozen-core approximation. Our calculatedphotoabsorption spectrum of magnesium dimer (Mg ) isomer is in excellent agreementwith the experiments both for peak positions, and intensities. Owing to the sufficientlyinclusive electron-correlation effects, these results can serve as benchmarks against whichfuture experiments, as well as calculations performed using other theoretical approaches,can be tested. ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . a t m - c l u s ] M a y . INTRODUCTION Clusters of group II elements, such as magnesium, are particularly interestingbecause they have two valence electrons, quasi-filled closed shells, and in bulk theyare metals. In the case of small clusters, the bonding between atoms is expectedto be of van der Walls type. This is evident in the case of extensively studiedmagnesium dimer. It exhibits a large bond length of 3.92 ˚A and 0.034 eV/atombinding energy. However, it is seen that for larger clusters this bonding becomesstronger. Thus, the study of divalent metals is appropriate for the evolution ofvarious cluster properties and to test various theoretical methods. Involvement ofmetal atoms in the clusters makes theoretical treatment a demanding task, mainlybecause of several nearly degenerate electronic states. In such situations, only multi-reference configuration interaction methods or coupled cluster singles doubles withperturbative triples (CCSD(T)) is known to provide best qualitative results [1]. Sincein this paper, we are dealing with small-sized clusters of magnesium, treated at alarge-scale multi-reference configuration interaction singles doubles level of theory,the results will be superior to other ab initio quantum chemical methods.There have been a large number of studies of equilibrium geometries and electronicstructure of small magnesium clusters [1–8]. Andrey et al. [2] studied the evolutionof the electronic structure of magnesium clusters with cluster size using all-electrondensity functional theoretical method. An evolution from non-metal to metal wasexplained using a gradient-corrected DFT calculations by Jellinek and Acioli [6],and by Akola et al [7]. Larger clusters were studied at DFT level by Kohn et al. [1]Kumar and Car performed ab initio density functional molecular dynamics studyof smaller magnesium clusters within local density approximation [3]. Stevens andKrauss calculated electronic structure of ground and excited states of Mg dimerusing multiconfigurational self-consistent field method [5]. Kaplan, Roszak, and2eszczynski investigated the nature of binding in the magnesium trimer at MP4level [8].The optical absorption in dimer was studied experimentally by McCaffrey andOzin [4], and Lauterwald and Rademann [9] in Ar, Kr and Xe matrices, while Bal-four and Douglas [10] measured it in the gas phase. Solov’yov et al. calculatedoptical absorption spectra of global minimum structures of magnesium clusters us-ing TDDFT and compared the spectra with results of classical Mie theory [11].However, to best of our knowledge, no other experimental or theoretical study existsfor optical absorption and excited states calculations of various low-lying isomersof magnesium clusters. The distinction of different isomers of a cluster has to bemade using experimental or theoretical techniques based upon properties which areshape and size dependent, unlike, mass spectroscopy which depends only on themass of the cluster. We have addressed this issue by performing large-scale cor-related calculations of optical absorption spectra of various isomers of magnesiumclusters Mg n (n=2–5), at MRSDCI level of theory. Hence, our theoretical resultscan help in distinguishing between different isomers of a cluster, when coupled withthe experimental measurements of their optical absorption. We also investigate thenature of optical excitations by analyzing the wavefunctions of various excited states.Furthermore, wherever possible, the results have been compared with the availableliterature. In earlier works, we reported similar calculations of optical absorptionspectra of various isomers of small boron and aluminum clusters [12, 13]. II. THEORETICAL AND COMPUTATIONAL DETAILS
A size-consistent coupled-cluster singles doubles (CCSD) level of theory along witha 6-311+G(d) basis set was used for geometry optimization, followed by vibrationalanalysis [14]. This basis set is well-suited for the ground state calculations. Different3 a) Mg , D ∞ h , Σ g (b) Mg , D h , A (cid:48) (c) Mg , D ∞ h , Π u (d) Mg ,C v , A (e) Mg , C v , B (f) Mg , T d , A (g) Mg , D h , B u (h) Mg ,D h , A g (i) Mg , D h , A (cid:48) (j) Mg , C v , A FIG. 1. Optimized geometries of Mg clusters considered in this work, along with thesymmetries of their ground state wave functions. Geometry optimization was carried outat the CCSD level, and all lengths are in ˚A units. spin multiplicities of the isomers were taken into account for the optimization todetermine the true ground state geometry. The process of optimization was initiatedby using the geometries reported by Lyalin et al. [2], based upon first principles DFT-based calculations. The final optimized geometries of the isomers are shown in Fig.1. For computing the optical absorption spectra, both ground and excited state wavefunctions for these optimized geometries were calculated using multireference singles-doubles configuration interaction (MRSDCI) method [15]. This approach consists of4enerating singly- and doubly-substituted configurations from a set of reference con-figurations, which are chosen based upon their contribution to the targeted wavefunctions obtained from a lower-level calculation based upon, say, single-referencesingles-doubles configuration interaction (SDCI) method. Optical absorption spec-tra are computed at each stage of the calculation, and the targeted wave functionsare analyzed to examine whether more reference configurations are needed. This pro-cedure is repeated until the absorption spectrum of the system under considerationconverges. Such an approach is equally efficient both for ground and excited statecalculations because it takes into account the electron correlation effects for all thetargeted states in an individualized manner, something which is not possible in singlereference approaches. The transition dipole moment matrix elements are calculatedusing these ground- and excited-state wavefunctions, and are subsequently utilized tocompute linear optical absorption spectrum assuming a Lorentzian line shape. Thenumerical approach described here has been extensively used in our earlier worksdealing with the optical properties of conjugated polymers [16–20], as well as atomicclusters [12, 13]. For the smallest cluster, namely, Mg dimer, it was possible to usethe full CI approach, within the frozen-core approximation.Since the computational effort involved in a CI calculation scales ≈ N , where N isthe total number of orbitals involved in the calculation, it can become intractable if alarge basis set, leading to a large number of molecular orbitals (MOs) is employed. Toreduce the MO basis set size, we employed the so-called “frozen-core approximation”,in which no virtual transitions are allowed from the chemical core of magnesiumatoms, thereby leading to two valence electrons per atom, which were treated asactive during the calculations. Furthermore, an upper limit of one hartree on theenergies of the virtual orbitals to be included in the calculations was imposed, so asto control the size of the CI expansion without compromising the accuracy of theoptical absorption spectrum. In the next section, we carefully examine the effects of5ll these approximations on our calculations.Further computational efficiency was achieved by making full use of point-groupsymmetries (D h , and its subgroups), wherever applicable. A. Choice of Basis Set
Electronic structure calculations generally depend upon the size and the qual-ity of basis set used. To explore the basis set dependence of computed spectra,we used several basis sets [21–23] to compute the optical absorption spectrum ofthe magnesium dimer. For the purpose, we used basis sets named aug-cc-pVDZ, cc-pVDZ, cc-pVTZ, 6-311++G(2d,2p), 6-311++G(d,p) and 6-311G(d,p), which consistof polarization functions along with diffuse exponents [21–23]. From the calculatedspectra presented in Fig. 2 the following trends emerge: the spectra computed byvarious correlation consistent basis sets (aug-cc-pVDZ, cc-pVDZ, cc-pVTZ) are ingood agreement with each other in the energy range up to 5 eV, while those obtainedusing the other basis sets (6-311++G(2d,2p), 6-311++G(d,p) and 6-311G(d,p)) dis-agree with them substantially, particularly in the higher energy range. Peaks at 5.6eV and 6.5 eV are seen only in the spectrum calculated using augmented basis set.Because of the fact that augmented basis sets are considered superior for molecularcalculations, we decided to perform calculations on the all the clusters using theaug-cc-pVDZ basis set.
B. Size of the CI Expansion
The electron correlation effects, both in ground state as well as excited states, weretaken into account in our calculations by the inclusion of relevant configurations inthe reference space of MRSDCI expansion. Larger the reference configuration space,6 I n t en s i t y ( a r b . un i t s ) Energy (eV) AUG-CC-PVDZCC-PVDZCC-PVTZ6-311++G(2d,2p)6-311++G(d,p)6-311G(d,p)
FIG. 2. Optical absorption in Mg calculated using various Gaussian contracted basis sets. larger will be the CI expansion, which is prohibitive for bigger systems. A goodchemical accuracy can be achieved by moderately sized CI expansions within theMRSDCI approach, provided the reference configurations are chosen carefully. InTable I we present the average number of reference states (N ref ) included in theMRSDCI expansion and the average number of configurations (N total ) for differentisomers. The average is computed over different irreducible representations requiredin the calculation of the ground and various excited states of a given isomer. Largescale nature of these calculations is obvious from the fact that the total numberof configurations in the CI expansion, N total , ranges from ≈ ABLE I. The average number of total configurations (N total ) involved in MRSDCI calcu-lations, ground state (GS) energies (in Hartree) at the MRSDCI level and relative energies(in eV) of various isomers of magnesium clusters.
Cluster Isomer N ref N total GS energy Relative(Ha) energy (eV)Mg (Fig. 1(a)) Linear 1 a (Fig. 1(b)) Equilateral Triangular 30 239465 -598.9270344 0.00(Fig. 1(c)) Linear 55 460187 -598.8759291 1.39(Fig. 1(d)) Isosceles Triangular-1 34 516337 -598.8569875 1.91(Fig. 1(e)) Isosceles Triangular-2 32 359780 -598.8093768 3.20Mg (Fig. 1(f)) Pyramidal 32 2962035 -798.5781385 0.00(Fig. 1(g)) Rhombus 29 1278632 -798.5405148 1.02(Fig. 1(h)) Square 35 1319301 -798.5278160 1.37Mg (Fig. 1(i)) Bipyramidal 11 3242198 -998.2044402 0.00(Fig. 1(j)) Pyramidal 28 2215749 -998.1980062 0.18 a Frozen core full configuration interaction calculation performed for Mg dimer. cluster (Mg ), to around three million for each symmetry subspace of Mg , implyingthat the electron-correlation effects have been adequately included.Before we discuss the absorption spectrum for each isomer, we present the groundstate energies along with the relative energies of each isomer are given in Table I. III. MRSDCI PHOTOABSORPTION SPECTRA OF MAGNESIUM CLUS-TERS
Next we present and discuss the results of our photoabsorption calculations foreach isomer. 8 . Mg The simplest cluster of magnesium is Mg with D ∞ h point group symmetry. Weobtained its CCSD optimized bond length to be 3.93 ˚A ( cf. Fig. 1(a)), which is inexcellent agreement with the experimental value 3.89 ˚A [10]. Using a DFT basedmethodology, several other theoretical values reported are in excellent agreementwith our optimized bond length of magnesium dimer, i.e., Kumar and Car reporteddimer bond length to be 3.88 ˚A [3] using density functional molecular dynamicswith simulated annealing, Janecek et al. computed bond length to be 3.70 ˚A [24]using DFT with LDA approximation, 3.8 ˚A bond length was reported by Stevensand Krauss using multiconfiguration self-consistent field approach [5], 3.91 ˚A bondlength of dimer was computed by Jellinek and Acioli using DFT with BP86 exchange-correlation functional [6], and Lyalin et al. reported 3.926 ˚A bond length using DFTwith B3LYP exchange-correlation functional [2].The computed photoabsorption spectra of Mg , as shown in Fig. 3, is character-ized by a couple of intense peaks in the 3 – 5 eV range, and by weaker absorptions,in between, and at higher energies. Table I of Supplementary Material presents themany-body wave functions of excited states contributing to various peaks [25]. Thefirst peak at 3.46 eV, due to the absorption of longitudinally polarized photons, isbecause of an excited state whose wave function is dominated by singly excited con-figuration H → L + 1, where symbols H and L denote HOMO and LUMO orbitalsrespectively. This peak is reported in the experimental photoabsorption at around3.36 eV [4, 10]. It is followed by a transversely polarized weaker absorption at 4.02eV, characterized by several singly excited configurations, including H → L + 8.The most intense peak occurs at 4.59 eV, whose wave function is also dominated bysingle excitations such as H − → L and H → L + 3. The location of this peak is inexcellent agreement with the experimental values of 4.59 eV reported by Lauterwald9 I n t e n s i t y ( a r b . un i t s ) Energy (eV)I || II ⊥ III ⊥ IV ⊥ V ⊥ FIG. 3. The linear optical absorption spectrum of Mg , calculated using the MRSDCIapproach. The subscript (cid:107) denotes the peak corresponding to the light polarization alongthe molecular axis, while the subscript ⊥ labels those polarized perpendiculars to it. Forplotting the spectrum, a uniform linewidth of 0.1 eV was used. and Rademann [9, 10], and 4.62 eV measured by McCaffrey and Ozin [4].The spectrum calculated using TDDFT by Solov’yov et al. [11] is in excellentagreement with our results. In their calculations, the first peak is seen at 3.3 eV,followed by the most intense peak at 4.6 eV. The overall photoabsorption profile isalso in accordance with our results. 10 . Mg We have optimized four low-lying geometries of magnesium trimer. The lowestenergy structure at CCSD optimized level has equilateral triangular shape with D h symmetry and bond lengths of 3.48 ˚A. This agrees well with other theoretical resultsreporting bond lengths, 3.51 ˚A [24], 3.48 ˚A [6], and 3.475 ˚A [2]. The next low-lying isomer of magnesium trimer has a linear structure, with D ∞ h symmetry. Theoptimized bond length is found to be 2.92 ˚A. The remaining two low-lying isomershave isosceles triangular shape, with C v point group symmetry. Not much has yetbeen reported on the bond lengths and electronic structure of these isomers.Figs. 4, 5, 6, and 7, present the photoabsorption spectra of these isomers, whileTables 2 – 5 of Supplementary Material [25] contain many-body wave functions ofimportant excited states contributing to various peaks.In the equilateral triangular isomer, the bulk of the oscillator strength is carried bya peak close to 3.75 eV. The linear isomer shows an altogether different absorptionspectrum with a number of peaks spread out in a wide energy range, with lightpolarized both parallel, and perpendicular, to the axis of the trimer. On the contrary,most of the oscillator strength in the absorption spectrum of isosceles triangularisomer-I is concentrated in the range of 3 – 5 eV. The lower-energy part of thespectrum of isosceles triangular isomer-II is somewhat red-shifted with respect tothe isosceles isomer-I, while, in the higher energy region, peaks are observed in theultraviolet range.The equilateral triangular isomer exhibits a weaker absorption peak at 2.6 eV,characterized by H → L and H → L + 4. This is followed by the most intensepeak at 3.7 eV due to the light polarized both parallel and perpendicular to theplane of the isomer and with a dominant contribution from excitations H → L , H → L + 2, and H − → L . This is confirmed by an experimental measurement of11hotoabsorption of Mg trimer in the argon matrix, which exhibits a peak at 3.64 eV[4]. Semi-major peaks at around 4.7 eV and 5.8 eV obtain dominant contributionfrom single excitations H → L + 7, H → L + 5, and H → L + 9. The latter peak isdue to photons polarized perpendicular to the plane of isomer.Comparing our results for the equilateral triangular isomer with the spectrumobtained by TDDFT calculations [11], we see very good agreement on the overallprofile of spectrum and excitation energies. The first peak is observed at 2.5 eV,followed by the most intense one at 3.7 eV, in the TDDFT spectrum.[11] Excitationenergies and relative oscillator strengths are also in good agreement with our results.Because the ground state of Mg linear isomer is a spin triplet, its many-particlewave function predominantly consists of a configuration with two degenerate singlyoccupied molecular orbitals, referred to as H and H in rest of the discussion. Thelinear trimer of magnesium cluster exhibits absorption in the entire energy range ex-plored. Very feeble peaks are observed at 0.9 eV and 2.3 eV, due to states dominatedby single excitations H → L + 8, H → L + 2, and H → L + 4. The wave function ofthe state leading to the second most intense peak at 2.9 eV is dominated by the con-figuration H → L + 3. The state leading to the most intense peak at 5.4 eV derivesalmost equal contributions from configurations H − → L and H − → L + 2. Theabsorption due to the longitudinally polarized light contributes to the lower energypart of the spectrum, while transversely polarized light contributes to the remaininghigher energy part of the spectrum.Both isosceles triangular isomers have a spin triplet ground state; hence theirexcited state wave functions will consist of configurations involving electronic exci-tations from singly occupied degenerate H and H molecular orbitals, in additionto other doubly occupied orbitals. In the case of isosceles triangular isomer - I ( cf .Fig. 6), the spectrum starts with a very feeble peak at 1.13 eV, leading to a statewhose wave function derives the main contribution from H → L + 1 configuration.12 I n t e n s i t y ( a r b . un i t s ) Energy (eV)I || II ||,⊥ III ||,⊥ IV || V ⊥ VI ⊥ VII ⊥ FIG. 4. The linear optical absorption spectrum of Mg equilateral triangle isomer, cal-culated using the MRSDCI approach. The peaks corresponding to the light polarized inthe molecular plane are labeled with the subscript (cid:107) , while the subscript ⊥ denotes thosepolarized perpendiculars to it. For plotting the spectrum, a uniform linewidth of 0.1 eVwas used. However, most of the absorption takes place in the energy range of 3 – 5 eV, withtwo equally intense peaks at 3.4 eV (peak V) and 4.2 (peak VII) eV, while the othertwo peaks (VI and IX) and a shoulder (VIII) in that range, are also quite intense.Peak V is due to three closely-spaced states, the first of which is reached by photonspolarized perpendicular to the plane of the triangle, while the other two are due tophotons polarized in the plane of the cluster. Wave functions of all the three statesderive dominant contributions from singly-excited configurations. The other most13 I n t e n s i t y ( a r b . un i t s ) Energy (eV)I ⊥ II || III || IV || V || VI ⊥ VII ⊥ VIII ⊥ IX ⊥ X ⊥ FIG. 5. The linear optical absorption spectrum of Mg linear isomer, calculated usingthe MRSDCI approach. The peaks corresponding to the light polarized along the molec-ular axis are labeled with the subscript (cid:107) , while the subscript ⊥ denotes those polarizedperpendiculars to it. For plotting the spectrum, a uniform linewidth of 0.1 eV was used. intense peak (VII) is due to two closely located states and displays mixed polariza-tion characteristics. Wave functions of both these excited states, in addition to thesingle excitations, derive important contributions from doubly excited configurationsas well.The absorption spectrum of the isosceles triangular isomer -II ( cf . Fig. 7) appearsred-shifted as compared to that of the previous isomer and exhibits a set of well-separated peaks. There is just one excited state contributing to the most intensepeak at 2.6 eV (peak V), which is due to the absorption of a photon polarized in14 I n t e n s i t y ( a r b . un i t s ) Energy (eV)I || II ||,⊥ III || IV || V ⊥,|| VI ||,⊥ VII || VIII ||,⊥ IX ||,⊥ FIG. 6. The linear optical absorption spectrum of Mg isosceles triangle isomer-I, cal-culated using the MRSDCI approach. The peaks corresponding to the light polarized inthe molecular plane are labeled with the subscript (cid:107) , while the subscript ⊥ denotes thosepolarized perpendiculars to it. For plotting the spectrum, a uniform linewidth of 0.1 eVwas used. the plane of the triangle. The wave function of this state mostly consists of theconfiguration H → L , with some contribution from a doubly-excited configuration H → L ; H → L + 10. Two almost equally intense peaks of absorption due toin-plane polarized photons occur at 3.5 eV (peak VII) and 3.9 eV (peak VIII). Oneexcited state each contributes to these peaks, and wave functions of these states aredominated by singly-excited configurations which include H − → H , H → L + 7,and H → L + 17. This isomer also exhibits a strong mixing of doubly excited15 I n t e n s i t y ( a r b . un i t s ) Energy (eV)I ⊥ II ⊥ III || IV || V || VI ⊥ VII || VIII || IX || X ||,⊥ XI ||,⊥ XII ||,⊥
XIII ⊥ FIG. 7. The linear optical absorption spectrum of Mg isosceles triangle isomer-II, cal-culated using the MRSDCI approach. The peaks corresponding to the light polarized inthe molecular plane are labeled with the subscript (cid:107) , while the subscript ⊥ denotes thosepolarized perpendiculars to it. For plotting the spectrum, a uniform linewidth of 0.1 eVwas used. configurations for excited states contributing to higher energy peaks. Significantdifferences in the optical absorption spectra of the two isosceles triangle shapedisomers point to a strong structure-property relationship when it comes to opticalproperties of these clusters. 16 . Mg The most stable isomer of Mg cluster has a closed-shell electronic ground state,with the structure of a perfect tetrahedron ( cf. Fig. 1(f)), corresponding to T d pointgroup symmetry, which, henceforth, we refer to as a pyramid. We computed theoptimized bond length to be 3.22 ˚A, which agrees well with the previously reportedvalues for this structure 3.09 ˚A [1], 3.33 ˚A [24], 3.18 ˚A [6], 3.31 ˚A [7], and 3.32 ˚A[8]. The rhombus isomer ( cf. Fig. 1(g)) with D h point group symmetry, and bondlength of 3.0 ˚A, along with the acute angle 63.5 ◦ , has B u electronic ground state,which is 1.02 eV higher than the global minimum structure. Square isomer ( cf. Fig.1(h)) with D h point group symmetry, and an optimized bond length of 3.06 ˚A, has A g electronic ground state, which is energetically 1.37 eV higher than the moststable structure.The absorption spectra of pyramidal, rhombus and square isomers are presented inFigs. 8, 9, and 10, respectively, while the many-particle wave functions of the excitedstates contributing to various peaks are presented in Tables 6, 7 and 8, respectively,of the Supplemental Material [25].Because of the three-dimensional structure of the pyramidal isomer, all threeCartesian components contribute to the transition dipole moments, thereby implyinga three-dimensional polarization of the incident photons with respect to the chosencoordinate system. The onset of absorption in this isomer occurs at 2.6 eV, due to astate whose many-particle wave function is dominated by configurations H − → L , H → L , and H − → L . The most intense peak in the spectrum is located at 4.54eV due to a state whose wave function is dominated by several single excitationssuch as H → L + 2, H − → L + 1, and H − → L + 3 etc. The TDDFTabsorption spectrum of this isomer reported by Solov’yov et al [11]. is slightly red-shifted compared our calculated spectrum, however, its absorption pattern is similar17 IG. 8. The linear optical absorption spectrum of pyramidal Mg isomer, calculatedusing the MRSDCI approach. The peaks corresponding to the light polarized along theCartesian axes are labeled accordingly. For plotting the spectrum, a uniform linewidth of0.1 eV was used. to ours in that a single most intense peak at 4.2 eV is followed by several less intensepeaks at higher energies.In the case of rhombus-shaped isomer, optical absorption starts with a weak peakat a rather low energy close to 1.00 eV, with the bulk of the oscillator strengthdistributed in the energy range 4 – 6 eV, consisting of several equally intense andclose-lying peaks. The first weak peak at 0.98 eV corresponds to a photon polarizedperpendicular to the plane of the molecule and is due to a state dominated by config-uration H → L + 1. The most intense peak at 4.7 eV is due to a photon polarized in18 I n t e n s i t y ( a r b . un i t s ) Energy (eV)I ⊥ II || III || IV ⊥ V || VI || VII || VIII ⊥ IX || X || XI ⊥ FIG. 9. The linear optical absorption spectrum of rhombus Mg , calculated using theMRSDCI approach. The peaks corresponding to the light polarized in the molecular planeare labeled with the subscript (cid:107) , while the subscript ⊥ denotes those polarized perpendic-ulars to it. For plotting the spectrum, a uniform linewidth of 0.1 eV was used. the plane of the isomer, and the corresponding excited state wave function is domi-nated by configurations H − → L + 1 and H − → L . It is preceded by a shoulderat 4.6 eV, with identical polarization properties, and an excited state wave functionwhich derives contributions from configurations H → L + 8 and H − → L . Themost intense peak corresponding to perpendicular polarization is located at 6.22 eV,with the excited state wave function dominated by doubly-excited configurations.The inversion symmetry of the ground state of the square isomer is just opposite tothat of the rhombus structure ( cf. Fig. 1), so that, as per dipole selection rule, the19xcited states contributing to the linear absorption spectra for the two structuresalso have opposite inversion symmetries. Quantitatively speaking, the absorptionspectrum of the square structure is slightly blue-shifted as compared to the rhombus,and red-shifted as compared to pyramidal isomer, with the majority of absorptionoccurring in the energy range 3–6 eV. The onset of absorption spectrum occurs at1.55 eV with a peak due to the light polarized in the plane of isomer, leading to astate whose wave function is a mixture predominantly of configurations H → L + 2, H → L + 10, and H → L + 15. This isomer, similar to the case of the rhombus,exhibits two very closely spaced high-intensity peaks, located at 4.50 eV and 4.73eV, both of which are due to the absorption of photons polarized in the plane of thecluster. The first of these peaks (peak V) is due to a state whose wave function isa mixture with almost equal contributions from single excitations as H − → L , H → L + 2, and H → L + 15, and also a doubly-excited configuration. The excitedstate causing the second one (peak VI) is dominated by singly excited configurations H → L + 20 and H → L + 24. The last peak of the computed spectrum (peak VII)has a relatively lower intensity and is due to a state dominated by single excitations H − → L + 13 and H → L + 20. D. Mg We optimized geometries of two isomers of Mg : (a) bipyramid with the D h symmetry and (b) a pyramidal structure with the C v point group symmetry. Thelowest lying bipyramidal isomer has A (cid:48) electronic ground state and is just 0.18 eVlower in energy as compared to the pyramid structure. Our optimized geometry forthe bipyramid has bond lengths of 3.15 ˚A and 3.52 ˚A, as against 3.00 ˚A, 3.33 ˚Areported by J. Jellinek and Acioli [6], and 3.09 ˚A, 3.44 ˚A reported by Andrey et al [2]. 20 I n t e n s i t y ( a r b . un i t s ) Energy (eV)I || II || III ||,⊥ IV || V || VI || VII || FIG. 10. The linear optical absorption spectrum of square Mg , calculated using the MRS-DCI approach. The peaks corresponding to the light polarized in the molecular plane arelabeled with the subscript (cid:107) , while the subscript ⊥ denotes those polarized perpendicularsto it. For plotting the spectrum, a uniform linewidth of 0.1 eV was used. The bipyramidal isomer of Mg cluster exhibits an absorption spectrum very dif-ferent from other isomers, as displayed in Fig. 11, while Table 9 of SupplementalMaterial [25] presents the many-particle wave functions of the excited states con-tributing to various peaks. The optical absorption spectrum of bipyramid Mg hasno absorption until 3.5 eV, while most of the absorption takes place in a narrow en-ergy range 5.3 – 6.3 eV. The absorption spectrum begins at 3.6 eV through a photonpolarized in the basal plane of the bipyramid, with a very feeble peak correspondingto a state whose wave function derives the main contribution from H − → L + 421onfiguration. This is followed by several such smaller peaks. The most intense peakat 5.4 eV has a dominant contribution from H → L + 1 along with other singlyexcited configurations, with absorption polarized again along the basal plane of thepyramid. A shoulder at 5.6 eV, however, corresponds to the absorption of light withpolarization along the z − direction, which is perpendicular to the basal plane. Thisfeature is caused by an excited state whose wave function is mainly a linear combi-nation of several singly-excited configurations. The TDDFT spectrum computed bySolov’yov et al [11]. shows optical activity in the energy range of 2–4 eV, which isnot observed in our calculated spectrum. However, a quasi-continuous spectrum isseen at higher energies in both calculations.The optical absorption spectrum of pyramid shaped isomer, to the best of ourknowledge, has not been computed, so far, by any other author. The entire absorp-tion spectrum of the pyramid shaped isomer is highly red-shifted as compared to thebipyramid isomer. The many-particle wave functions of excited states contributingto the peaks are presented in Table 10 of Supplemental Material [25]. The opticalabsorption of this isomer exhibits a few feeble peaks in the low energy range, withthe onset of spectrum occurring at 2.24 eV through photons polarized in the basalplane of the pyramid, as well as perpendicular to it. The wave functions of the twoexcited states contributing to this peak are dominated by configurations H − → L and H → L + 2. The second most intense peak close to 3.5 eV is well separated fromthe most intense one located at 4.2 eV. The former has mixed polarization character-istics, with two states dominated by configurations H − → L + 3 and H − → L ,besides several other single excitations. The most intense peak is due to light polar-ized perpendicular to the basal plane of the pyramid, and wave function of the excitedstate involved is dominated by configurations H − → L + 1 and H − → L + 3.The last absorption peak in the probed energy range is located at 6.27 eV, causedby a photon polarized perpendicular to the base of the pyramid, and is due to an22 IG. 11. The linear optical absorption spectrum of bipyramidal Mg isomer, calculatedusing the MRSDCI approach. The peaks corresponding to the light polarized along theCartesian axes are labeled accordingly. For plotting the spectrum, a uniform linewidth of0.1 eV was used. excited state deriving the main contribution from a doubly-excited configuration,along with several single excitations. Pyramid shaped isomer exhibits prominentoptical absorption in the higher energy range, with almost regularly spaced peaksof declining intensities, in contrast to single major peak observed in the spectrumof bipyramidal isomer. These differences can help in experimental identification ofgeometries of various isomers, through optical absorption spectroscopy.23 I n t e n s i t y ( a r b . un i t s ) Energy (eV)I ||,⊥ II || III ||,⊥ IV || V ⊥ VI || VII ||,⊥
VIII || IX ⊥ X || XI ⊥ FIG. 12. The linear optical absorption spectrum of pyramidal Mg , calculated using theMRSDCI approach. The peaks corresponding to the light polarized along the base of thepyramid are labeled with the subscript (cid:107) , while the subscript ⊥ denotes those polarizedperpendiculars to it. For plotting the spectrum, a uniform linewidth of 0.1 eV was used. IV. CONCLUSIONS AND OUTLOOK
In this work, large-scale first-principles electron correlated calculations of photoab-sorption spectra of several low-lying isomers of magnesium clusters Mg n , (n=2–5),were presented. For the case of magnesium dimer, we employed one of the bestpossible electronic structure methods, namely FCI method, within the frozen-coreapproximation, to compute its electronic states. Calculations for the remaining clus-ters were performed using MRSDCI approach, which takes excellent account of the24lectron-correlation effects both for ground and excited states. We have also ana-lyzed the nature of the many-particle wave functions of the excited states visiblein the absorption spectra. Distinct signature spectra are exhibited by isomers ofa given cluster, suggesting a strong structure-property relationship. This behaviorcan be utilized in the experiments to distinguish between different isomers of a clus-ter, using optical absorption spectroscopy. Given the fact electron-correlation effectswere included in our calculations in a sophisticated manner by means of large-scaleCI expansions, we believe that our results can be used as theoretical benchmarks ofabsorption spectra of Mg clusters, against which both the experimental and othertheoretical results can be compared. We hope that our work will lead to experimen-tal measurements of the optical absorption spectra of magnesium clusters of variousshapes and sizes. ACKNOWLEDGMENTS
R.S. acknowledges the Council of Scientific and Industrial Research (CSIR) andScience and Engineering Research Board (SERB) India, for research fellowships(09/087/(0600)2010-EMR-I), (PDF/2015/000466). Authors kindly acknowledgecomputational resources provided by National Param Yuva Supercomputing Facil-ity, C-DAC, Pune. [1] A. Kohn, F. Weigend, and R. Ahlrichs, Phys. Chem. Chem. Phys. , 711 (2001).[2] A. Lyalin, I. A. Solov’yov, A. V. Solov’yov, and W. Greiner, Phys. Rev. A , 063203(2003).[3] V. Kumar and R. Car, Phys. Rev. B , 8243 (1991).
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