Introduction

The ground state terms (spin and angular momenta) of isolated atoms are determined by Hund’s rules1, which are explained by the lowering of the electronuclear attraction energy2,3,4,5. For molecules and clusters, such term rules do not exist. Group theory allows us to determine the possible 2S+1Ξ molecular terms, where S denotes the total spin and Ξ the symmetry species, but there is no systematic way to figure out which of these is the ground state. Direct experimental observation or quantum chemical total energy calculation is only available for a few prototype systems.

Hund’s first rule of maximum spin multiplicity holds for many organic molecules6,7,8, but not all9. Diatomic molecules on the other hand, tend to have spin singlet ground states with the exception of O2 and B2 (see e.g. refs 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 for diatomic molecules of main group elements). 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. Moreover, recent attempts to generalize Hund’s rules for molecules8,20 or clusters21,22,23 only focus on a spin multiplicity rule and trends or rules for Ξ (symmetry) terms remain completely uncharted territory.

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. For example, strained quasi-1D chains of Al may exhibit ferromagnetism24,25 and Aln clusters with even n = 2, 4, 6, 8 have spin-triplet ground states21,26,27,28,29. Al3 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 unresolved28,30,31,32,33. The present benchmark study confirms that Al2 has the 3Πu high-spin ground state and unambiguously shows that Al3 has the low-spin ground state. The Al2 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. For Al3, 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. Moreover, fear each spin state, we find a simple rule for the Ξ terms. The Ξ term with least node wavefunction is most stable and for terms with equal number of nodes, the one with most bonds is most stable. Notice that for diatomic molecules, Ξ is the angular momentum along the internuclear axis, which can be either minimized or maximized by this rule.

Results

Al2 has five stationary states, , 1Πu, 1Δg, 3Πu and , corresponding to the occupation of different molecular orbitals by two 3p electrons. Al3 has three stationary states, , 4A2 and 4B1, corresponding to the occupation of different molecular orbitals by three 3p electrons. Their equilibrium nuclear geometries and corresponding total energies E are shown in Table 1. Hartree-Fock (HF) calculation predicts Al2 and Al3 to have 3Πu and 4A2 high-spin ground states, respectively. Inclusion of Coulomb correlation by CAS-SCF (see Methods) maintains the high-spin ground state of Al2, but stabilizes the low-spin ground state of Al3. At the same time, high-spin terms of Al2 (3Πu, ) and Al3 (4A2, 4B1) become nearly degenerate; the energy difference between them is smaller than 0.01 a.u. The 3Πu ground state for Al2 is consistent with experiment34 and our prediction of the ground state for Al3 is corroborated by the Stern-Gerlach experiment30. More importantly, the ground state of Al2 is consistent with both Hund’s first and second rules, whereas Al3 violates both of them.

Table 1 Total energies and equilibrium structures of Al2 and Al3.
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Potential energy components

Traditionally, Hund’s rules have been interpreted as an energy gain due to the inter-electron repulsion potential energy Vee35,36,37 and more recently as an energy gain due to the electronuclear attraction Ven2,4,5,7,8. In order to analyze whether or not similar energy lowering mechanisms can be invoked for Al2 and Al3, we decompose the total energies given in Table 1 into potential energy components shown in Fig. 1. In both HF (dashed lines) and CAS-SCF (solid lines) calculations for each stationary state of Al2 and Al3, repulsion terms Vee (red lines) and Vnn (blue lines; inter-nuclear repulsion) are positive and the attraction term Ven (purple lines) is negative. The total energies E of Al2 and Al3 calculated by CAS-SCF always lie lower than those calculated by HF. For both Al2 and Al3, upon inclusion of Coulomb correlation by CAS(6, 26) and CAS(9, 18), respectively, the individual potential energy components Ven, Vee and Vnn composing V change as follows: both Vee and Vnn increase and Ven decreases. The correlation energies Ec = ECAS − EHF, along with Vc, , and , defined similarly, are unique to each molecular term; Ec < 0 always and for the components we find , and .

Figure 1
figure 1

Energy components for each equilibrium state of Al2 and Al3.

HF, CAS(6, 26) for Al2 and CAS(9, 18) for Al3 levels are dashed lines and solid lines, respectively. E (black), V (orange), Vee (red), Vnn (blue) and Ven (purple) are given in hartree atomic units.

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For Al2, both HF and CAS-SCF predict

The correlation energies, however, exhibit the different trend

making the excitation energies smaller. For Al3, HF predicts

and CAS-SCF predicts

i.e. the level ordering is altered by correlation effects. For correlation energies we find

For both Al2 and Al3, the strongest correlation effect, i.e. greatest correlation energy Ec, is observed for the 1Δg and low-spin terms, respectively. For Al3, this correlation effect is strong enough to alter the level ordering of the molecular terms, but for Al2 not. Thus, the relative stability of the Al2 molecular terms can be discussed based on Fermi correlation (Pauli’s exclusion principle) and HF calculations, but for Al3, Coulomb correlation included by CAS-SCF is crucial for the description of molecular terms.

For Al2, Hund’s first and second rules predict

which is valid only for the spin-triplet terms. The spin-singlet terms exhibit an opposite trend to Hund’s second rule. Term stabilities have earlier been interpreted by either Vee35,36,37 or Ven2,4,5,7,8, which imply that total energy differences are dominated by one potential energy component Vi(i = ee, en, or nn), i.e., the total energy should follow the trend of this dominant Vi. Figure 1, however, shows that

which is different to Eq. (1). Here the + sign corresponds to i = en and the − sign to i = ee and i = nn. Clearly total energy trends do not follow any one particular potential energy component. Although the highest spin multiplicity (Hund’s first) rule does not follow any of the potential energy components Ven, Vee and Vnn, the Ξ terms, when observed for spin-triplet and spin-singlet states individually, exhibit the following trends

Note that Eq. (8a) has the opposite sign convention to Eqs (7) and (8b). 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!

Fermi correlation and bond structure

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 Al2, inclusion of Coulomb correlation via CAS-SCF does not alter the relative stability of the Al2 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.e., the nodal structure of the wavefunction. Al2 has 3pσg and 3pπu bonding orbitals and for the spin-singlet and spin-triplet terms, the most stable Ξ term has an occupied 3pσg orbital, i.e., the least node configuration. The stability of the spin-triplet 3Πu against the spin-singlet term also follows from HF theory. Starting from the nodeless wavefunction, moving one electron from the 3pσg into a 3pπu with parallel spin (forming the 3Πu term) lowers the total energy in three steps: (i) for fixed orbitals and Al–Al bond length, Vee is lowered for spin parallel electrons35; (ii) relaxing the electronic orbitals lowers the total energy further; and (iii) relaxing the Al–Al bond length lowers the total energy further still. Repeating steps (ii) and (iii) obviously keeps lowering the total energy until convergence is found; these steps can be roughly associated to changes in Ven and Vnn, respectively, but as seen in Fig. 1, for Al2 Vee and Vnn actually increase despite the initial lowering of Vee in step (i). For the Ξ terms we find that for a given spin multiplicity, the total energy increases as the number of nodes in the wavefunction increases.

Table 2 Electronic configurations of valence electrons of Al2 and Al3.
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Coulomb correlation and bond structure

The above discussion fails for Al3. Inclusion of Coulomb correlation via CAS-SCF un-stabilizes the spin-quadruplet terms despite their possession of two electrons in 3pσ type orbitals (a1 and b2 for 4A2 and two a1s for 4B1) on the Al3 molecular plane. We analyze the effects of Coulomb correlation based on the electron density distribution change defined by ρc = ρCAS − ρHF, where ρCAS and ρHF are the total electron densities calculated by CAS-SCF and HF, respectively. The crucial Coulomb correlation that alters the Al3 term stabilities occurs at CAS(9, 12) and therefore we evaluate ρCAS for Al2 and Al3 using CAS(6, 18) and CAS(9, 12), respectively. The ρc shown in Figs 2 and 3 for Al2 and Al3, respectively, are evaluated at the equilibrium nuclear configurations obtained by CAS(6, 18) and CAS(9, 12), respectively.

Figure 2
figure 2

Schematic of bonding orbitals (a) and Coulomb correlation induced electron density distribution change ρc for stationary states of Al2 (b–e).Here ρc = ρCAS − ρHF, where ρCAS is evaluated at CAS(6, 18).

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Figure 3
figure 3

Schematic of bonding orbitals (a) and Coulomb correlation induced electron density distribution change ρc for stationary states of Al3 (b–d). Here ρCAS is evaluated at CAS(9, 12).

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Al2

The Coulomb correlation effects are analyzed based on the bonding 3pσg and 3pπu orbitals shown in panel (a) of Fig. 2. Panels (b)–(e) of Fig. 2 show the electron density differences ρc for the 3Πu, , 1Πu and terms in the planes P1 and P2 corresponding to the 3pσg and 3pπu bonding orbitals. The blue areas indicate a depletion of electron density and the yellow–orange–red areas an increase of electron density. For the terms, these P1 and P2 planes are equivalent. The ρc analysis is omitted for the 1Δg term, which is not correctly represented in the HF calculation.

The CAS(6, 18) calculation includes various configurations including up to 3d orbitals, but the essence of the Coulomb correlation effects can be described based on the mixing of the bonding 3pσg and 3pπu orbitals. As shown in panels (b)–(e) of Fig. 2, the electron density distribution corresponding to orbitals occupied in HF theory (Table 2) is depleted and increases corresponding to bonding orbitals not occupied in HF theory. For the 3Πu, 1Πu and terms that in HF have an occupied 3pσg orbital, there is a depletion in ρc along the bond axis and for the that in HF does not have an occupied 3pσg orbital, there is an increase. Likewise, ρc is negative in the regions corresponding to 3pπu orbitals occupied in HF theory and positive in the regions where the 3pπu orbitals are not occupied in HF theory. 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.

Al3

Al3 has the low-spin ground state, against expectations from Hund’s first rule or the spin-state stabilization mechanism for Al2 pertaining to HF theory. Thus, the energy lowering effect of Coulomb correlations is different for the low-spin term and the high-spin 4A2 and 4B1 terms. The effect of these Coulomb correlations is discussed based on the 3pσ and 3pπ orbitals shown in panel (a) of Fig. 3. Panels (b)–(d) of Fig. 3 show the electron density differences ρc for the , 4A2 and 4B1 terms in the plane P1 of the nuclei of Al3 and its perpendicular plane P2, which is a reflection symmetry plane of Al3. Notice that the nuclei of the spin-doublet term form equilateral triangle, whereas the spin-quadruplet terms 4A2 and 4B1 correspond to isosceles triangles. Ensuingly, the bonding orbitals for the low-spin and high-spin terms are quite different.

has a doubly occupied bonding orbital, a singly occupied orbital and a doubly degenerate e′ LUMO. The orbital is a π bond where the plane of nuclei is a nodal plane and the a1′ is a σ bond with the charge density lobe in the center of the triangle. Both and orbitals have C3v symmetry, resulting in an equilateral trimer with D3h symmetry. The doubly degenerate e′ LUMO corresponds to a σ bond with charge density lobes at all three sides of the triangle. The main Coulomb correlation effect is similar to what was discussed above for Al2. There is a depletion of electron in the regions corresponding to the and orbitals occupied in HF theory and an increase in the region corresponding to the e′ orbitals, as seen in Fig. 3(b).

For the spin-quadruplet terms one of the electrons occupies either one of the e′ orbitals. Individually these orbitals have the C2v symmetry, yielding Jahn-Teller distorted isosceles triangles as described in Table 2. This changes the also symmetry species of the occupied and orbitals into b1 and a1, respectively, but these orbitals still maintain their nature as π and σ bonds with similar charge density lobes as described above for the equilateral triangle. The newly formed a1 or b2 orbitals for the 4B1 or 4A2 terms, have charge density lobes at the base or legs of the triangle, respectively, as shown in Fig. 3(a). The LUMO of the 4B1 and 4A2 terms are b2 and a1, respectively, i.e., the other one of the e′ orbitals for an equilateral triangle. For the spin-quadruplet terms, the main Coulomb correlation effect is the mixing of the a1 or b2 orbitals, which can be seen Fig. 3(c,d) as a depletion of electron density along the legs (base) of the triangle for 4A2 (4B1) and the corresponding along the base (legs) of the triangle.

Because of different symmetries, the Coulomb correlations for the low-spin and the high-spin terms of Al3 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 configurations2 in total stabilizes the Al3 low-spin ground state. Thus, both Hund’s first rule and the mechanism that stabilizes the high-spin ground state of Al2 are violated because the breaking of symmetry of the Al3 spin-quadruplet configurations reduces their Coulomb correlation. Note that Hund’s maximum spin multiplicity rule is violated under exactly the opposite conditions as postulated by Kutzelnigg and Morgan20.

Larger clusters

Application of the term rules described above for other clusters is straight forward. We illustrate this generalization by predicting ground state terms for Al4 and Al5. For both clusters, we consider previously described planar and pyramidal structures26,28,38; incidentally, our discussion below offers a new interpretation for why planar geometries are favored against pyramidal ones39. We predict 3B1u and 2B1 ground states for Al4 and Al5, respectively, well in agreement with previous works21,26,28. The structures and spin multiplicities agree also with density-functional calculations29,38, which however give no information of the symmetry species Ξ.

Al4

Al4 can have spin-singlet, spin-triplet and spin-quintet states due to different configurations of four 3p electrons, shown in Table 3. The HOMO of any of the Al4 terms with pyramidal structure (3-fold degenerate 2t1 orbitals) do not form σ-type bonds, so for any spin multiplicity, the least node wavefunctions corresponds to a planar geometry. The planar Al4 spin-quintet terms (high spin) always have at least one occupied antibonding molecular orbital, such as 2b3u, 2b2u, 2b3g and 2b2g, whereas the spin-triplet and singlet terms 3B1u, 3Au, 3B1g and 1Ag have valence electrons occupying in bonding orbitals (1b1g, 1b1u and 3ag). Thus, Fermi correlation stabilizes the spin-triplet terms with possession of most-occupied σ-type bonding orbitals, i.e., 3B1u state. 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 3B1u and 1Ag states. As seen in stability of Al2’s spin triplet terms, such Coulomb correlation effect cannot reverse the relative stability for 3B1u and 1Ag and thus we predict 3B1u as the ground state term of Al4.

Table 3 Electronic configurations of valence electrons for planar and pyramidal geometries for Al4.
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Al5

Al5 can have spin multiplicities up to spin-sextet due to different configurations of five 3p-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. The planar spin-quadruplet terms (intermediate spin state) only have partial bonds, such as 3b1, 4b1 and 2b2, which leaves only spin-doublet terms with strong σ-type bonds. Thus for Al5, we predict the planar 2B1 spin-doublet term, which has the least node structure ground state.

Table 4 Electronic configurations of valence electrons for planar and pyramidal geometries for Al5.
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Discussion

Our benchmark first principles calculation predicts the 3Πu (high-spin) and (low-spin) ground states for Al2 and Al3, respectively. Detailed analysis of potential energy components of the total energy reveal that previous interpretations, attributing atomic or molecular term stabilization to either Vee35,36 or Ven2,5,8 are, in general, not valid for multi-atom systems. The relative stability of the Ξ terms for a given spin multiplicity for either Al2 and Al3 follows simple arguments based on bonding structures: For a given spin multiplicity the Ξ term possessing the most-occupied σ bonding orbitals (least node structure) is stabilized within the one-electron orbital picture according to Hartree-Fock (HF) theory. In addition, HF theory tends to stabilize the high-spin term due to Fermi correlation (Pauli exclusion principle). Coulomb correlation lowers the energy by mixing some of the orbitals occupied in HF theory with nearby unoccupied orbitals. For Al2, the Coulomb correlation effects are similar for all terms, but for Al3, Coulomb correlation alters the relative term stability. For Al3, 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 Al3. 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. We demonstrate this predictive power by predicting 3B1u and 2B1 ground states for Al4 and Al5, respectively.

Methods

The total energy E of the 2S+1Ξg/u term of an Aln cluster in the Born-Oppenheimer approximation is given by , where is a many-electron wavefunction and the operators give the electron kinetic energy, the inter-nuclear repulsion, the electronuclear attraction and the inter-electron repulsion, respectively. The expectation values for each operator , henceforth denoted as O(2S+1Ξg/u), are calculated using the GAMESS package40. We use both Hartree-Fock (HF) method and complete active space self-consistent field (CAS-SCF) method. Our CAS-SCF many-electron wavefunctions contain configuration interactions among the 3s and 3p valence shells and empty 3d-derived orbitals: CAS(6, 26) and CAS(9, 18) for Al2 and Al3, respectively. CAS(n, m) stands for a CAS-SCF calculation with n active spaces and m active electrons. Atomic orbitals are expanded within the aug-cc-pVTZ basis set and all nuclear positions are relaxed. This gives a virial ratio of −V/T = 2.00000 ± 0.00003 for each molecular term 2S+1Ξg/u.

Additional Information

How to cite this article: Yoshida, D. and Raebiger, H. Term rules for simple metal clusters. Sci. Rep. 5, 15760; doi: 10.1038/srep15760 (2015).