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Ab initio predictions of optically allowed transitions in Na20. Nature of excitations and influence of geometry
Volume 2 13, number $6
CHEMICAL PHYSICS LETTERS
15 October 1993
Ab initio predictions of optically allowed transitions in NazO.
Nature of excitations and influence of geometry V. BonaWKouteclj a, P. Fantucci b, C. Fuchs a, C. Gatti ‘, J. Pittner a and S. Polezzo ’ * Institutfiir Physikalische und Theoretische Chemie, Freie UniversitiitBerlin. Takustrasse3, D-14195Berlin, Germany b Dipartimento di Chimica horganica. Metallorganica e Analitica, Centro de1Consiglio NazionaIe delle Ricerche, Universitadi Milano, 20133 Milano, Italy c Centro Consiglio Nazionale delle Ricerche ‘Relazioni Strutturae Reattivita Chimica’: Via Goigi 19,20133 Milano, Italy Received 1 July 1993; in final form 12 August 1993
An ab initio investigation shows that the deformed section of the fee lattice for Nazowith Td symmetry gives rise to three intense transitions with differing oscillator strengths due to excitations localized in different parts of the cluster. These predictions are in agreement with the experimental findings by two-photon femtoseeond spectroscopy and the depletion technique. The study of different structures for Nalo reveals that the spectroscopic patterns strongly depend on the position of the nuclei.
1. Introduction Alkali metal clusters with 8 and 20 valence electrons have attracted considerable attention since it was suggested that they are particularly stable and that they assume a spherical shape in the framework of the jellium model [ 11. They also possess specific optical properties characterized by large absorption cross sections in relatively narrow energy intervals [2-61. These features parallel the shell closing and giant resonances found for nuclei in spite of the different nature of the forces involved. The question has been raised as to whether the closed-shell clusters differ from open shell ones not only in their stabilities but also in their spectroscopic patterns and whether their observed properties are mainly due to their geometry and/or the number of valence electrons. It has been shown that a comparison of depletion spectra [ 3-61 with the ab initio predictions of transition energies (T,) and oscillator strengths UC)for the most stable geometries of small alkali metal clusters (Nan=3_8, Nagf, LinZ3_*) with even and odd number of valence electrons allows a geometrical assignment and an interpretation of the spectroscopic patterns [7-l 5 1. The agreement between the predicted and measured observables has been remark522
ably good in spite of the fact that the theoretical treatment is valid for T=O K, while the exact temperature of the experiments is not known. However, the hypothesis has been adopted that with increasing cluster size the temperature might also increase so that several energetically close-lying isomers might be responsible for the observed spectra. Therefore, different model approaches, classical [ 21 and quantum mechanical [ 16-l 81, neglecting the positions of nuclei have frequently been used to estimate the spectroscopic properties of alkali metal clusters. However, it was also explicitly shown [ 181 that the distribution of oscillator strengths for closed-shell clusters strongly depends on the choice of the positive background which is the major approximation in the jellium model. An interpretation of the depletion spectrum of NaZO based on ab initio predictions which respect the positions of the nuclei needed additional experimental confirmation (cf. ref. [ 71). In view of new experimental developments in the framework of two-photon femtosecond (fs) spectroscopy [ 191, the ab initio study of excited states of Nazo gained in importance, since the lifetimes of the resonances are now available and the effect of temperature seems to be less critical. In fact, two-photon fs experiments on Naa [ 19,201 yield four resonances with different
0009-2614/93/% 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 213, number 5.6
CHEMICAL PHYSICS LETTERS
lifetimes lying in the energy interval of the dominant band obtained by the depletion technique [3,4]. These new experimental findings fully confirm our ab initio configuration interaction (CI) predictions obtained for the lowest energy Td structure of Na, and support our molecular interpretation [ 9,101. In this Letter we present ab initio random phase approximation (RPA) results for three different structures of NaZOwith tetrahedral (Td) symmetry. The comparison between the predicted spectrum for the lowest-energy structure, which is a deformed section of the fee lattice, and the two-photon fs spectrum [ 191 allows an interpretation and assignment of the three recorded resonances.
2. Results and discussion As in previous studies of excited states of Na,=,_, [ 10,131, the calculations were carried out using an ab initio effective core potential corrected for core-valence correlation (ECP-CVC) with the atomic orbital basis set (4s,4p) contracted to three s functions and one p function. An SCF geometry optimization was carried out only for a limited number of topologies under different constraints (with a relatively small number of variables), including topologies of tetrahedral symmetry and structures built from pentagonal subunits (cf. ref. [ 2 1 ] for more details). Large scale multireference configuration interaction (MR-CI) calculations were performed for the SCF “optimized” structures in order to determine their energy ordering at a higher level of theory. Since a large number of excited states had to be calculated for 20 valence electrons, methods which are designed for a simultaneous determination of many excited states and transition moments such as RPA or multiconfigurational linear response (MCLR) are more practicable than the CI procedure (for theoretical aspects of calculations of excited states cf. refs. [ 7,221). According to our experience, the ab initio RPA yields relatively reliable results for highly symmetrical structures owing to the high degeneracy of the one-electron levels, which reinforces a dominance of single excitations [ 7,23 1. As is well known, RPA accounts for double excitations only in an approximate manner. However, in order to verify the reliability of the results, we per-
15October1993
formed MCLR calculations for the lowest-energy structure of Nazo, allowing for single, double and higher-order excitations within a chosen set of active orbitals for a given number of electrons. Since the RPA results for T, andf, do not substantially deviate from those obtained by the MCLR method [ 2 1] we present the RPA spectra for all three Td structures in fig. 1. The emphasis here is put on the broad interpretation of the results rather than on the details, which will be published elsewhere [ 211. Of the structures studied, the lowest energy form is a “relaxed fee” geometry (structure I in fig. 1). It was obtained from the “exact” fee structure (structure II), which is a section of the fee lattice, characterized by constant internuclear distances, by optimization of three distances and one angle, and it is lower in energy by 0.16 eV with respect to this exact fee geometry, at both SCF and MR-CI levels of computation. The third tetrahedral structure (III) studied exhibits a more “spherical” shape and, at the MRCI level, has 0.89 eV higher energy than the relaxed fee form (I). All three Td structures are characterized by the presence of three groups of atoms forming (i) an inner tetrahedron, (ii) an outer tetrahedron and (iii) twelve symmetry equivalent atoms distributed on a sphere (fig. 1) . Although all three structures have Td symmetry, their corresponding spectroscopic patterns differ substantially as shown in fig. 1. The “spherical” structure (III) gives rise to a single dominant transition located at 2.45 eV with f,z 10.59, while three intense transitions at zz2.2, 2.43 and z 2.8 eV have been calculated for both fee geometries (structures I and II). The relaxation introduced in the rigid fee form does not appreciably affect the locations of the transitions but does considerably change the values of the oscillator strengths for the first two intense transitions with T, = 2.2 and 2.43 eV (cf. fig. 1). Fig. 1 illustrates qualitatively the reasons for these differences, which are related to relatively small changes in the positions of the nuclei. The one-electron levels involved in the leading single excitations which are responsible for the intense transitions are localized (predominantly but not entirely) in one of the three different “shells” of the cluster. From fig. 1 it can be seen that, for the intense transitions, only a few configurations have large coefficients. These involve single excitations of 12 electrons from the three sets 523
CHEMICAL PHYSICS LETTERS
Volume 213, number 5,6
structureI
15 October 1993
(A&cF = 0.00, A&x = 0.00)’
state
Important single excitations with Jci( > 0.25
61Tz
0.69(le --+ ltl)
-0.49(2t,
7’TZ
0.42(2tz --t 4tz)
o-o -036(le -t 412)
o-o 0.31(‘& + 5t2)
o-o -0.30(%2 +
0-e
3.872
3.529
11’Tz
3.708
--+ 412)
-0.3qzq
+ 4q
e-0 0.31(2al -
4t2)
e-0 401)
O-0
0.53(212 * 4W)
-0.41(212 --t 5t2)
o-0
0.35(2q
O-0
-
3t2)
O-0
The ground state configuration is 11$ It; 2af 21: le*
NazO fee (T,)
II structure II
71T4
(A&F
= 0.16, A&, = 0.16)l
state
Important single excitations with lcil > 0.25
6’T*
0.52(le +
422)
o-o 0.31(2al + 3tl)
1 1’Ta 50
f,
0.37(le +
It,)
o-
-0.31(‘&
--) 1tJ
o-o 0.58(2t2 -
o-0.39(2q
-) 3t*)
0-a
.o 0.0
--) 4t2)
o-o
O-0 0.61@~ + 4f2)
4q)
-0.37(2tz
-0.35(2tz
O-0
-t 5tl)
O-0
The ground state configuration is la: ltii 2af 2tz le’
NazO
lT,,) III 23l Tp,
Structure III
(AEscr
= 1.41, A&r = 0.89)’
state
Important single excitations with jci( > 0.25
1O’Tz
0.34(212 + 111) 0.33(&
)
23lT2
o-o 0.85(le -
-
o-o 0.28(2i2 -
4fl)
-O.Sl(zal
512)
o-o -0.27(2t2 + 30,)
o-o
-
4t2)
O-0
712)
o-o The ground state configuration is la: ltf 2a: le’ zti 1 Relative energies with respect to the energetically lowed structure I (in eV) Fig. 1. Optically allowed transitions with oscillator strengths for the three Td structures of Nal,,. The three groups of atoms labeled by white, black and shaded circles belong to the inner tetrahedron, the outer tetrahedron and to the twelve-atom shell, respectively.
524
Volume 2 13, number 5,6
15 October 1993
CHEMICAL PHYSICS LETTERS
of highest occupied one-electron levels 2ai, 2t, and le to the five sets of unoccupied MOs: 3tz, It,, 4t2, 4at (or 3at ) and 5tz (cf. also fig. 30 of ref. [ 7 ] ). For the fee structures I and II, the 2a, and le, 2t, MOs are mainly localized at the atoms of the outer and inner tetrahedra, respectively. For the relaxed fee structure (I) the most intense transition at 2.43 eV is dominated by excitations localized either within the inner tetrahedron or between the inner and the outer tetrahedron. The red-shifted transition to the 6 ‘T2 state with lower intensity also involves excitations from the inner tetrahedron to the twelve-atom shell. In contrast, excitations from the inner to the outer tetrahedron, or within the latter, represent the largest contribution to the 11 ‘T, transition, which is blue shifted with respect to the dominant one. The composition of the excited state for this transition (11 ‘T2) remains almost unchanged for the exact section of the fee lattice (structure II) and, consequently, so does the value of fe. The largest change in the nature of the leading excitations occurs in the transition to the 7 ‘Tr state of structure II for which the atoms belonging to the outer tetrahedron are no longer involved and the correspondingf, value is diminished by more than 5OW.Notice that, owing to the high symmetry of the Td structures, all three directions of polarization axes are identical, and therefore the three intense transitions cannot be attributed to geometric deformations with respect to the spherical structure, as assumed in the spheroidal jellium model. Fig. 1 also shows that the three intense transitions of the relaxed fee lattice have several common leading configurations although their contributions differ in magnitude and sign, From the analysis of the single excitations responsible for the weak and strong transitions [ 2 1] the conclusion can be drawn that the different intensities arise from interference among a small number of leading contigurations (cf. also ref. [ 71). The sums of the calculated oscillator strengths for the transitions in the energy interval up to 4.2 eV are 17.67, 17.65 and 16.19 for structures I, II and III, respectively. The finding with two-photon fs spectroscopy of three resonances located at z 2.19, 2.4 1 and 2.76 eV [ 191 supports our prediction of the spectroscopic pattern for the deformed fee Td structure of NalO,for which the lowest energy has been obtained (cf. fig. 2). (It should be noted that the calculated transition
0.0 350.9
4cQ.o
450.0
5OQ.o
wavelength
550.0
600.0
65
(nm)
I,.,.,,.,,,,....,...,,,,..., 400
:::q
500
600
Na20 Td 1 Structure I
400
500
600
Fig. 2. Comparison of predicted transitions and oscillator strengths for the deformed section of the fee lattice (structure I) with experimental spectra of ref. [ 5] (window) and [ 191 (middle). The measurements of ref. [ 21 are denoted by 0 in the upper part.
at x 1.75 eV= z 708 nm (fig. 1) cannot be assigned to an experimentally observed band since, to our knowledge, no measurements have been performed in this energy region.) The predicted spectrum of structure I is also in agreement with the depletion spectrum [ 51, since the two intense transitions lo525
Volume 213, number 5,6
CHEMICAL PHYSICS LETTERS
cated at z 2.2 and 2.4 eV lie in the energy region of the dominant band and the third one at x2.8 eV corresponds to the shoulder (cf. fig. 2). However, other Nazo isomers of comparably high stability might be involved in thermal equilibrium because of the relatively high temperature of the depletion experiments (which is assumed to be at least 300 K), and therefore could contribute to the observed spectrum. Other relatively stable structures with lower symmetry than the Td forms, obtained from a molecular dynamics study [ 24 1, exhibit different spectroscopic patterns and will be reported elsewhere [ 2 I].
15October 1993
Nazionale delle Ricerche (CNR). We thank Professor G. Gerber for communicating his results prior to publication and Professor J. Kouteckjr for useful discussions.
References [ I ] W.A. de Heer, W.D. Knight, M.Y. Chou and M.L. Cohen, Solid State Phys. 40 (1987) 93.
[ 21 W. de Heer, K. Selby, V. Kresin, J. Masui, M. Vollmer, A. Chatelain and W.D. Knight, Phys. Rev. Letters 59 (1987)
1805. [ 31 C. Wang, S. Pollack, D. Cameron and M.M. Kappes, J. Chem. Phys. 93 (1990) 3787.
[ 41 C. Wang, S. Pollack, T. Dahlseid, G.M. Koretsky and M.M. 3. Summary
Kappes, J. Chem. Phys. 96 ( 1992) 7931.
[ 5] S. Pollack, C. Wang and M.M. Kappes, I. Chem. Phys. 94 We have shown that not only the overall symmetry but also the precise positions of the nuclei are decisive for the spectroscopic pattern of Nazo. Although it cannot be excluded that the most stable structure of Na2,, has not yet been found, the deformed section of the fee lattice is presently the lowest-energy form which gives rise to a spectroscopic pattern in satisfactory agreement with experimental findings. The calculated spectrum is characterized by three intense transitions which arise from excitations mainly localized in different parts of the cluster. The involvement of almost all twenty electrons giving rise to one intense transition has not been confirmed. Even in the case of the “spherical” structure III, the value offe for the most intense transition is considerably smaller cf ex10.59) than 20. Additional calculations show that even for this type of structure the exact positions of the nuclei are important for the spectroscopic pattern, indicating that approximations introduced in the potential can easily influence the predictions in an arbitrary manner. In spite of the delocalized nature of bonding in alkali metal clusters the evidence for collective excitations of the surface plasmon type has not been found for small clusters such as Nalo, although it may be present in larger systems.
Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft (Sfh 337, Energy and Charge Transfer in Molecular Aggregates) and the Consi& 526
(1991) 2496.
[ 61 C. Brechignac, P. Cahuzac, F. Carlier, M. de Frutos and J. Leygnier, Z. Physik D 19 ( 1991) 1.
[ 7] V. BonaEiOKouteck$,P. Fantucci and J. Koutecki, Chem. Rev. 91 (1991) 1035.
[ 81V. BonaEiC-Kouteckj,P. Fantucci and 1. Koutecu, Chem. Phys. Letters 166 (1990)32. [ 91 V. BonaEiC-Koutecw, M.M. Kappes, P. Fantucci and J. Koutecu, Chem. Phys. Letters 170 ( 1990) 26. [ 1O] V. Bona%-Koutecq, P. Fantucci and J. Kouteck$, J. Chem. Phys. 93 (1990) 3802. [ 11 ] V. BonaEiCKouteckji,P. Fantucci and J. Kouteckjr, Chem. Phys.Letters 146 (1988) 518. [ 121V. BonaEiC-Kouteck$,J. Gaus, M.F. Guest and J. Koutece, J. Chem. Phys. 96 (1992) 4934. [ 131V. Bona&C-Kouteckj, J. Pittner, C. Scheuch, M.F. Guest and J. Koutecky, J. Chem. Phys. 96 (1992) 7938. [ 141Ph. Dugourd, J. Blanc, V. Bona&C-Kouteclj, M. Broyer, J. Chevaleyre, J. Koutcck$,J. Pittner, J.-P. Wolf and L. W&e, Phys. Rev. Letters 67 (1991) 2638. [ 15] J. Blanc, V. Bona&C-Kouteckg,M. Broyer, J. Chevaleyre, Ph. Dugourd, J. KoutecQ, C. Scheuch, J.-P. Wolf and L. WBste,J. Chem. Phys. 96 ( 1992) 1793. [ 161W. Ekardt, 2. Penzar and M. SunjiC,Phys. Rev. B 33 (1986) 3702. [ 171C. Yannouleas, R.A. Broglia, M. Brack and P.F. Bortignon, Phys. Rev. Letters 63 ( 1989) 255. [ 18] C. Guet and W. R. Johnson, Phys. Rev. B 45 (1992) 11283. [ 191R. Thalweiser, C. Rothenfusser, V. Weiss and G. Gerber, to be published. [20] T. Baumert, R. Thalweiser, V. Weiss and G. Gerber, Z. Physik D, in press. [21] V. BonaEiC-Koutccw, P. Fantucci, C. Fuchs, C. Gatti, J. Pittner, S. Polezzo and .I. Koutecw, to be published. [22] C. Fuchs, V. Bona&Kouteckjr and J. Kouteckjr, J. Chem. Phys. 98 (1990) 3121. [23] C. Gatti, S. Polezzo and P. Fantucci, Chem. Phys. Letters 175 (1990) 645. [24]U. Rethlisberger and W. Andreoni, J. Chem. Phys. 94 (1991) 8129.
CHEMICAL PHYSICS LETTERS
15 October 1993
Ab initio predictions of optically allowed transitions in NazO.
Nature of excitations and influence of geometry V. BonaWKouteclj a, P. Fantucci b, C. Fuchs a, C. Gatti ‘, J. Pittner a and S. Polezzo ’ * Institutfiir Physikalische und Theoretische Chemie, Freie UniversitiitBerlin. Takustrasse3, D-14195Berlin, Germany b Dipartimento di Chimica horganica. Metallorganica e Analitica, Centro de1Consiglio NazionaIe delle Ricerche, Universitadi Milano, 20133 Milano, Italy c Centro Consiglio Nazionale delle Ricerche ‘Relazioni Strutturae Reattivita Chimica’: Via Goigi 19,20133 Milano, Italy Received 1 July 1993; in final form 12 August 1993
An ab initio investigation shows that the deformed section of the fee lattice for Nazowith Td symmetry gives rise to three intense transitions with differing oscillator strengths due to excitations localized in different parts of the cluster. These predictions are in agreement with the experimental findings by two-photon femtoseeond spectroscopy and the depletion technique. The study of different structures for Nalo reveals that the spectroscopic patterns strongly depend on the position of the nuclei.
1. Introduction Alkali metal clusters with 8 and 20 valence electrons have attracted considerable attention since it was suggested that they are particularly stable and that they assume a spherical shape in the framework of the jellium model [ 11. They also possess specific optical properties characterized by large absorption cross sections in relatively narrow energy intervals [2-61. These features parallel the shell closing and giant resonances found for nuclei in spite of the different nature of the forces involved. The question has been raised as to whether the closed-shell clusters differ from open shell ones not only in their stabilities but also in their spectroscopic patterns and whether their observed properties are mainly due to their geometry and/or the number of valence electrons. It has been shown that a comparison of depletion spectra [ 3-61 with the ab initio predictions of transition energies (T,) and oscillator strengths UC)for the most stable geometries of small alkali metal clusters (Nan=3_8, Nagf, LinZ3_*) with even and odd number of valence electrons allows a geometrical assignment and an interpretation of the spectroscopic patterns [7-l 5 1. The agreement between the predicted and measured observables has been remark522
ably good in spite of the fact that the theoretical treatment is valid for T=O K, while the exact temperature of the experiments is not known. However, the hypothesis has been adopted that with increasing cluster size the temperature might also increase so that several energetically close-lying isomers might be responsible for the observed spectra. Therefore, different model approaches, classical [ 21 and quantum mechanical [ 16-l 81, neglecting the positions of nuclei have frequently been used to estimate the spectroscopic properties of alkali metal clusters. However, it was also explicitly shown [ 181 that the distribution of oscillator strengths for closed-shell clusters strongly depends on the choice of the positive background which is the major approximation in the jellium model. An interpretation of the depletion spectrum of NaZO based on ab initio predictions which respect the positions of the nuclei needed additional experimental confirmation (cf. ref. [ 71). In view of new experimental developments in the framework of two-photon femtosecond (fs) spectroscopy [ 191, the ab initio study of excited states of Nazo gained in importance, since the lifetimes of the resonances are now available and the effect of temperature seems to be less critical. In fact, two-photon fs experiments on Naa [ 19,201 yield four resonances with different
0009-2614/93/% 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 213, number 5.6
CHEMICAL PHYSICS LETTERS
lifetimes lying in the energy interval of the dominant band obtained by the depletion technique [3,4]. These new experimental findings fully confirm our ab initio configuration interaction (CI) predictions obtained for the lowest energy Td structure of Na, and support our molecular interpretation [ 9,101. In this Letter we present ab initio random phase approximation (RPA) results for three different structures of NaZOwith tetrahedral (Td) symmetry. The comparison between the predicted spectrum for the lowest-energy structure, which is a deformed section of the fee lattice, and the two-photon fs spectrum [ 191 allows an interpretation and assignment of the three recorded resonances.
2. Results and discussion As in previous studies of excited states of Na,=,_, [ 10,131, the calculations were carried out using an ab initio effective core potential corrected for core-valence correlation (ECP-CVC) with the atomic orbital basis set (4s,4p) contracted to three s functions and one p function. An SCF geometry optimization was carried out only for a limited number of topologies under different constraints (with a relatively small number of variables), including topologies of tetrahedral symmetry and structures built from pentagonal subunits (cf. ref. [ 2 1 ] for more details). Large scale multireference configuration interaction (MR-CI) calculations were performed for the SCF “optimized” structures in order to determine their energy ordering at a higher level of theory. Since a large number of excited states had to be calculated for 20 valence electrons, methods which are designed for a simultaneous determination of many excited states and transition moments such as RPA or multiconfigurational linear response (MCLR) are more practicable than the CI procedure (for theoretical aspects of calculations of excited states cf. refs. [ 7,221). According to our experience, the ab initio RPA yields relatively reliable results for highly symmetrical structures owing to the high degeneracy of the one-electron levels, which reinforces a dominance of single excitations [ 7,23 1. As is well known, RPA accounts for double excitations only in an approximate manner. However, in order to verify the reliability of the results, we per-
15October1993
formed MCLR calculations for the lowest-energy structure of Nazo, allowing for single, double and higher-order excitations within a chosen set of active orbitals for a given number of electrons. Since the RPA results for T, andf, do not substantially deviate from those obtained by the MCLR method [ 2 1] we present the RPA spectra for all three Td structures in fig. 1. The emphasis here is put on the broad interpretation of the results rather than on the details, which will be published elsewhere [ 211. Of the structures studied, the lowest energy form is a “relaxed fee” geometry (structure I in fig. 1). It was obtained from the “exact” fee structure (structure II), which is a section of the fee lattice, characterized by constant internuclear distances, by optimization of three distances and one angle, and it is lower in energy by 0.16 eV with respect to this exact fee geometry, at both SCF and MR-CI levels of computation. The third tetrahedral structure (III) studied exhibits a more “spherical” shape and, at the MRCI level, has 0.89 eV higher energy than the relaxed fee form (I). All three Td structures are characterized by the presence of three groups of atoms forming (i) an inner tetrahedron, (ii) an outer tetrahedron and (iii) twelve symmetry equivalent atoms distributed on a sphere (fig. 1) . Although all three structures have Td symmetry, their corresponding spectroscopic patterns differ substantially as shown in fig. 1. The “spherical” structure (III) gives rise to a single dominant transition located at 2.45 eV with f,z 10.59, while three intense transitions at zz2.2, 2.43 and z 2.8 eV have been calculated for both fee geometries (structures I and II). The relaxation introduced in the rigid fee form does not appreciably affect the locations of the transitions but does considerably change the values of the oscillator strengths for the first two intense transitions with T, = 2.2 and 2.43 eV (cf. fig. 1). Fig. 1 illustrates qualitatively the reasons for these differences, which are related to relatively small changes in the positions of the nuclei. The one-electron levels involved in the leading single excitations which are responsible for the intense transitions are localized (predominantly but not entirely) in one of the three different “shells” of the cluster. From fig. 1 it can be seen that, for the intense transitions, only a few configurations have large coefficients. These involve single excitations of 12 electrons from the three sets 523
CHEMICAL PHYSICS LETTERS
Volume 213, number 5,6
structureI
15 October 1993
(A&cF = 0.00, A&x = 0.00)’
state
Important single excitations with Jci( > 0.25
61Tz
0.69(le --+ ltl)
-0.49(2t,
7’TZ
0.42(2tz --t 4tz)
o-o -036(le -t 412)
o-o 0.31(‘& + 5t2)
o-o -0.30(%2 +
0-e
3.872
3.529
11’Tz
3.708
--+ 412)
-0.3qzq
+ 4q
e-0 0.31(2al -
4t2)
e-0 401)
O-0
0.53(212 * 4W)
-0.41(212 --t 5t2)
o-0
0.35(2q
O-0
-
3t2)
O-0
The ground state configuration is 11$ It; 2af 21: le*
NazO fee (T,)
II structure II
71T4
(A&F
= 0.16, A&, = 0.16)l
state
Important single excitations with lcil > 0.25
6’T*
0.52(le +
422)
o-o 0.31(2al + 3tl)
1 1’Ta 50
f,
0.37(le +
It,)
o-
-0.31(‘&
--) 1tJ
o-o 0.58(2t2 -
o-0.39(2q
-) 3t*)
0-a
.o 0.0
--) 4t2)
o-o
O-0 0.61@~ + 4f2)
4q)
-0.37(2tz
-0.35(2tz
O-0
-t 5tl)
O-0
The ground state configuration is la: ltii 2af 2tz le’
NazO
lT,,) III 23l Tp,
Structure III
(AEscr
= 1.41, A&r = 0.89)’
state
Important single excitations with jci( > 0.25
1O’Tz
0.34(212 + 111) 0.33(&
)
23lT2
o-o 0.85(le -
-
o-o 0.28(2i2 -
4fl)
-O.Sl(zal
512)
o-o -0.27(2t2 + 30,)
o-o
-
4t2)
O-0
712)
o-o The ground state configuration is la: ltf 2a: le’ zti 1 Relative energies with respect to the energetically lowed structure I (in eV) Fig. 1. Optically allowed transitions with oscillator strengths for the three Td structures of Nal,,. The three groups of atoms labeled by white, black and shaded circles belong to the inner tetrahedron, the outer tetrahedron and to the twelve-atom shell, respectively.
524
Volume 2 13, number 5,6
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CHEMICAL PHYSICS LETTERS
of highest occupied one-electron levels 2ai, 2t, and le to the five sets of unoccupied MOs: 3tz, It,, 4t2, 4at (or 3at ) and 5tz (cf. also fig. 30 of ref. [ 7 ] ). For the fee structures I and II, the 2a, and le, 2t, MOs are mainly localized at the atoms of the outer and inner tetrahedra, respectively. For the relaxed fee structure (I) the most intense transition at 2.43 eV is dominated by excitations localized either within the inner tetrahedron or between the inner and the outer tetrahedron. The red-shifted transition to the 6 ‘T2 state with lower intensity also involves excitations from the inner tetrahedron to the twelve-atom shell. In contrast, excitations from the inner to the outer tetrahedron, or within the latter, represent the largest contribution to the 11 ‘T, transition, which is blue shifted with respect to the dominant one. The composition of the excited state for this transition (11 ‘T2) remains almost unchanged for the exact section of the fee lattice (structure II) and, consequently, so does the value of fe. The largest change in the nature of the leading excitations occurs in the transition to the 7 ‘Tr state of structure II for which the atoms belonging to the outer tetrahedron are no longer involved and the correspondingf, value is diminished by more than 5OW.Notice that, owing to the high symmetry of the Td structures, all three directions of polarization axes are identical, and therefore the three intense transitions cannot be attributed to geometric deformations with respect to the spherical structure, as assumed in the spheroidal jellium model. Fig. 1 also shows that the three intense transitions of the relaxed fee lattice have several common leading configurations although their contributions differ in magnitude and sign, From the analysis of the single excitations responsible for the weak and strong transitions [ 2 1] the conclusion can be drawn that the different intensities arise from interference among a small number of leading contigurations (cf. also ref. [ 71). The sums of the calculated oscillator strengths for the transitions in the energy interval up to 4.2 eV are 17.67, 17.65 and 16.19 for structures I, II and III, respectively. The finding with two-photon fs spectroscopy of three resonances located at z 2.19, 2.4 1 and 2.76 eV [ 191 supports our prediction of the spectroscopic pattern for the deformed fee Td structure of NalO,for which the lowest energy has been obtained (cf. fig. 2). (It should be noted that the calculated transition
0.0 350.9
4cQ.o
450.0
5OQ.o
wavelength
550.0
600.0
65
(nm)
I,.,.,,.,,,,....,...,,,,..., 400
:::q
500
600
Na20 Td 1 Structure I
400
500
600
Fig. 2. Comparison of predicted transitions and oscillator strengths for the deformed section of the fee lattice (structure I) with experimental spectra of ref. [ 5] (window) and [ 191 (middle). The measurements of ref. [ 21 are denoted by 0 in the upper part.
at x 1.75 eV= z 708 nm (fig. 1) cannot be assigned to an experimentally observed band since, to our knowledge, no measurements have been performed in this energy region.) The predicted spectrum of structure I is also in agreement with the depletion spectrum [ 51, since the two intense transitions lo525
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CHEMICAL PHYSICS LETTERS
cated at z 2.2 and 2.4 eV lie in the energy region of the dominant band and the third one at x2.8 eV corresponds to the shoulder (cf. fig. 2). However, other Nazo isomers of comparably high stability might be involved in thermal equilibrium because of the relatively high temperature of the depletion experiments (which is assumed to be at least 300 K), and therefore could contribute to the observed spectrum. Other relatively stable structures with lower symmetry than the Td forms, obtained from a molecular dynamics study [ 24 1, exhibit different spectroscopic patterns and will be reported elsewhere [ 2 I].
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Nazionale delle Ricerche (CNR). We thank Professor G. Gerber for communicating his results prior to publication and Professor J. Kouteckjr for useful discussions.
References [ I ] W.A. de Heer, W.D. Knight, M.Y. Chou and M.L. Cohen, Solid State Phys. 40 (1987) 93.
[ 21 W. de Heer, K. Selby, V. Kresin, J. Masui, M. Vollmer, A. Chatelain and W.D. Knight, Phys. Rev. Letters 59 (1987)
1805. [ 31 C. Wang, S. Pollack, D. Cameron and M.M. Kappes, J. Chem. Phys. 93 (1990) 3787.
[ 41 C. Wang, S. Pollack, T. Dahlseid, G.M. Koretsky and M.M. 3. Summary
Kappes, J. Chem. Phys. 96 ( 1992) 7931.
[ 5] S. Pollack, C. Wang and M.M. Kappes, I. Chem. Phys. 94 We have shown that not only the overall symmetry but also the precise positions of the nuclei are decisive for the spectroscopic pattern of Nazo. Although it cannot be excluded that the most stable structure of Na2,, has not yet been found, the deformed section of the fee lattice is presently the lowest-energy form which gives rise to a spectroscopic pattern in satisfactory agreement with experimental findings. The calculated spectrum is characterized by three intense transitions which arise from excitations mainly localized in different parts of the cluster. The involvement of almost all twenty electrons giving rise to one intense transition has not been confirmed. Even in the case of the “spherical” structure III, the value offe for the most intense transition is considerably smaller cf ex10.59) than 20. Additional calculations show that even for this type of structure the exact positions of the nuclei are important for the spectroscopic pattern, indicating that approximations introduced in the potential can easily influence the predictions in an arbitrary manner. In spite of the delocalized nature of bonding in alkali metal clusters the evidence for collective excitations of the surface plasmon type has not been found for small clusters such as Nalo, although it may be present in larger systems.
Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft (Sfh 337, Energy and Charge Transfer in Molecular Aggregates) and the Consi& 526
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