E, No. Dekker, Morokhov I D i dr. Nauk ; Morokhov I D et al. D 3 Hagena O F Z. Press, Huisken F Adv. Nauk 1 ; Smirnov B M Sov. Nauk ; Eletskii A V Phys. Nauk ; Smirnov B M Phys. A Herschbach D Rev. Nauk ; Smalley R E Rev. Nauk ; Curl R F Rev. Nauk ; Kroto H W Rev. A 53 Ditmire T et al. A 57 Zweiback J et al. Plasmas 9 Eletskii A V Usp. R Campargue Berlin: Springer, p. Nauk ; Makarov G N Phys. A Springate E et al.

A 61 Auguste T et al. B 20 McPherson A et al. Nature Dobosz S et al. A 56 R Ditmire T et al.

## Growth mechanisms and stability of atomic clusters | QuantBioLab

Kvantovaya Elektron. Laser Part. Beams 21 73 Fukuda I i dr. Nature Zweiback J et al. A Madison K W J. B 20 Campbell E E B et al. Nauk 11 ; Leonas V B Sov. A 44 R Vandenbosch R et al. B 88 Fortov V E et al. Impact Eng. D 9 29 Yasumatsu H et al. For molecules and clusters, such term rules do not exist. Direct experimental observation or quantum chemical total energy calculation is only available for a few prototype systems.

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Diatomic molecules on the other hand, tend to have spin singlet ground states with the exception of O 2 and B 2 see e. These molecules tend to have ground states that minimize the internuclear bond length, which may be associated with a lowering of the electronuclear attraction energy, but whether or not such discussion generalizes to metallic clusters remains unknown. Simple Al clusters emerge as the ideal model system to study term rules. Bulk Al is paramagnetic, but in low dimensional structures Al atoms may spontaneously align their spins. Al 3 on the other hand has spin-doublet low-spin and spin-quadruplet high-spin configurations, but which one of these is the ground state remains unresolved 28 , 30 , 31 , 32 , The Al 2 high-spin state is stabilized by Fermi correlation, which is not overcome by Coulomb correlation that tends to increase the stability of low-spin terms.

## Term rules for simple metal clusters

For Al 3 , however, the high-spin term has a symmetry broken geometry that preempts effective Coulomb correlation from taking place, thus un-stabilizing the high-spin term. Such symmetry lowering can debilitate high-spin terms of any multi-atom system. Al 3 has three stationary states, , 4 A 2 and 4 B 1 , corresponding to the occupation of different molecular orbitals by three 3 p electrons. Their equilibrium nuclear geometries and corresponding total energies E are shown in Table 1. In order to analyze whether or not similar energy lowering mechanisms can be invoked for Al 2 and Al 3 , we decompose the total energies given in Table 1 into potential energy components shown in Fig.

In both HF dashed lines and CAS-SCF solid lines calculations for each stationary state of Al 2 and Al 3 , repulsion terms V ee red lines and V nn blue lines; inter-nuclear repulsion are positive and the attraction term V en purple lines is negative. E black , V orange , V ee red , V nn blue , and V en purple are given in hartree atomic units. For both Al 2 and Al 3 , the strongest correlation effect, i. For Al 3 , this correlation effect is strong enough to alter the level ordering of the molecular terms, but for Al 2 not.

Figure 1 , however, shows that. Clearly total energy trends do not follow any one particular potential energy component. Note that Eq. Thus, for a given spin multiplicity, the potential energy components follow the same trend as total energies, but the sign may vary case by case! Since clear term rules cannot be described based on the individual energy components discussed above, we turn our attention to the bond structures given in Table 2 for each molecular term.

For Al 2 , inclusion of Coulomb correlation via CAS-SCF does not alter the relative stability of the Al 2 terms, so the relative term stabilities can be understood purely based on Fermi correlation Pauli exclusion principle. This leads to a simple description based on the bond structures of the different terms, i. Repeating steps ii and iii obviously keeps lowering the total energy until convergence is found; these steps can be roughly associated to changes in V en and V nn , respectively, but as seen in Fig.

The above discussion fails for Al 3. Panels b — e of Fig. The blue areas indicate a depletion of electron density, and the yellow—orange—red areas an increase of electron density. For the terms, these P 1 and P 2 planes are equivalent.

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As shown in panels b — e of Fig. Because all these Coulomb correlation effects essentially occur among the same set of orbitals, all of which are bonding, the effects are similar. Because the Coulomb correlation effects are similar for all terms, Coulomb correlation does not alter the relative stability of them, and the discussion above of term stability based on Fermi correlations and wavefunction nodal structure is sufficient.

Thus, the energy lowering effect of Coulomb correlations is different for the low-spin term and the high-spin 4 A 2 and 4 B 1 terms. Panels b — d of Fig. Notice that the nuclei of the spin-doublet term form equilateral triangle, whereas the spin-quadruplet terms 4 A 2 and 4 B 1 correspond to isosceles triangles. Ensuingly, the bonding orbitals for the low-spin and high-spin terms are quite different. Both and orbitals have C 3 v symmetry, resulting in an equilateral trimer with D 3 h symmetry.

The main Coulomb correlation effect is similar to what was discussed above for Al 2. Individually these orbitals have the C 2 v symmetry, yielding Jahn-Teller distorted isosceles triangles as described in Table 2. The newly formed a 1 or b 2 orbitals for the 4 B 1 or 4 A 2 terms, have charge density lobes at the base or legs of the triangle, respectively, as shown in Fig. For the spin-quadruplet terms, the main Coulomb correlation effect is the mixing of the a 1 or b 2 orbitals, which can be seen Fig.

Because of different symmetries, the Coulomb correlations for the low-spin and the high-spin terms of Al 3 are fundamentally different. For the spin-doublet term, the main Coulomb correlation is the mixing of two occupied states and an unoccupied doubly degenerate state, whereas for the spin-quadruplet terms, the main Coulomb correlation is due to the mixing of one occupied and one unoccupied state.

Coulomb correlation acts strongly among states nearby in energy and real space, and for the spin-quadruplet terms, the Jahn-Teller distortion imposes a severe limitation on the availability of such nearby states for mixing. This, combined with the fact that Coulomb correlation even without geometrical distortions is larger for low-spin configurations 2 in total stabilizes the Al 3 low-spin ground state. Application of the term rules described above for other clusters is straight forward. We illustrate this generalization by predicting ground state terms for Al 4 and Al 5.

For both clusters, we consider previously described planar and pyramidal structures 26 , 28 , 38 ; incidentally, our discussion below offers a new interpretation for why planar geometries are favored against pyramidal ones We predict 3 B 1 u and 2 B 1 ground states for Al 4 and Al 5 , respectively, well in agreement with previous works 21 , 26 , Al 4 can have spin-singlet, spin-triplet, and spin-quintet states due to different configurations of four 3 p electrons, shown in Table 3.

The planar Al 4 spin-quintet terms high spin always have at least one occupied antibonding molecular orbital, such as 2 b 3 u , 2 b 2 u , 2 b 3 g , and 2 b 2 g , whereas the spin-triplet and singlet terms 3 B 1 u , 3 A u , 3 B 1 g , and 1 A g have valence electrons occupying in bonding orbitals 1 b 1 g , 1 b 1 u , and 3 a g.

Coulomb correlation, which enhances the electron density on the nodal plane of HOMO s , makes Al-Al bonds on the molecular plane stronger for both 3 B 1 u and 1 A g states. Al 5 can have spin multiplicities up to spin-sextet due to different configurations of five 3 p -electrons. All pyramidal, and the planar spin-sextet terms have at least one electron in antibonding or nonbonding orbitals see Table 4 , and thus cannot be more stable than the planar spin-doublet or spin-quadruplet terms.

Thus for Al 5 , we predict the planar 2 B 1 spin-doublet term, which has the least node structure ground state. Detailed analysis of potential energy components of the total energy reveal that previous interpretations, attributing atomic or molecular term stabilization to either V ee 35 , 36 or V en 2 , 5 , 8 are, in general, not valid for multi-atom systems. In addition, HF theory tends to stabilize the high-spin term due to Fermi correlation Pauli exclusion principle.

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Coulomb correlation lowers the energy by mixing some of the orbitals occupied in HF theory with nearby unoccupied orbitals. For Al 2 , the Coulomb correlation effects are similar for all terms, but for Al 3 , Coulomb correlation alters the relative term stability. For Al 3 , breaking of symmetry of the the spin-quadruplet terms significantly limits the orbital mixing and energy lowering by Coulomb correlation. The high symmetry of the spin-doublet term, on the other hand, allows for mixing with degenerate levels followed by a much larger energy lowering by Coulomb correlation, stabilizing the low-spin ground state of Al 3.

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## Atoms, Molecules, and Clusters: Structure, Reactivity, and Dynamics

These stabilization mechanisms are not specific for Al clusters, and serve as simple term rules to determine the ground state of arbitrary multi-atomic systems. Related Meeting. Conference Program. Sunday pm - pm Arrival and Check-in.

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Samuel Leutwyler University of Bern, Switzerland. Junichi Nishitani Kyoto University, Japan. Steffen Spieler University of Innsbruck, Austria. Richard Bowles University of Saskatchewan, Canada. Hanna Vehkamaki University of Helsinki, Finland. Paul Scheier University of Innsbruck, Austria. Thomas Taxer University of Innsbruck, Austria. Gary Douberly.