1 Three and FourMembered Atomic Rings
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Orbitals in Inorganic Chemistry
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D 3h
D 4h
Oh
Scheme 1 Symmetries of metal clusters for (r, t, p) approach
Hückel
Hückel
Möbius
Hückel
s
pr
pt
pπ
Hückel
Hückel
Hückel
Hückel
s
pρ
pτ
pπ
s
pρ
pτ
Scheme 2 Hückel or Möbius conjugation of the arrays of sorbitals and radial (pr), tangential
(pt), and perpendicular (pp) porbitals
(Scheme 3). The qualitative energy levels (Scheme 4) show the number of valence
electrons necessary to obtain closedshell electronic structures. Each orbital in
the sorbital set is assumed to be occupied by a pair of electrons since the sorbital
energies are low and separate from those of the porbital ones, especially for
heavy atoms. The total number of valence electrons for the closedshell structures
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S. Inagaki
a
b
c
Hückel
Möbius
Hückel
Scheme 3 Energy splittings of three and fourorbital arrays
a
b
X
X
18
16
14
12
10
8
X
X
18
16
22
20
10
12
14
8
6
4
6
4
2
2
s
pρ
pτ
pπ
Threemembered ring
s
pρ
pτ
pπ
Fourmembered ring
Scheme 4 Orbital energy levels
is 4N+ 2 for the threemembered rings and 8N+ 6 for the fourmembered rings
[7]. The valence electron rules have been supported by ab initio calculations
(Tables 1 and 2) [7].
2.1.1 4N Valence Electron rule
4N valence electrons do not allow regular polygons (D3h and D4h) as the ground
states of trimers and tetramers [7].
There are no exceptions in Tables 1 and 2 [7]. For 4N electron systems, the singlet ground states of trimers and tetramers do not assume three and fourmembered
ring structures of D3h and D4h symmetry, respectively.
Orbitals in Inorganic Chemistry
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Table 1 Ground state geometries of trimersa
Number of valence electrons
4N+ 2
10
14
Al 3 (D3h)
Ga3 (D3h)
Si
Ge
P
As

4N
(D3h)
(D3h)
(D3h)
(D3h)
2–
3
2–
3
+
3
+
3
18
8
S3 (C2v)
Se3 (D3h)
Al (D∞h)
Ga (D∞h)
+
3
+
3
12
16
20
Si3 (C2v)
Ge3 (C2v)
P3 (D∞h)
As3 (D∞h)
S32+ (C2v)
Se32+ (C2v)
S32 (C2v)
Se32 (C2v)

Calculated at the UB3LYP/631 + G(d) level [7]
a
Table 2 Ground state geometries of tetramersa
Number of valence electrons
4N + 2
4N
8N+ 2
10
18
8N + 6
26
Al42+ (D∞h) Si42– (D2d) S42– (C2)
Ga42+ (D∞h) Ge42– (D2d) Se42– (C2)
P42+ (D2d)
As42+ (D2d)
12
16
20
24
14
22
Si42+ (D4h)
Ge42+ (D4h)
Al42 (D4h)
Ga42 (D4h)
P42– (D4h) Al4 (C2h) Si4 (D2h) P4 (Td) S4 (D2d)
As42– (D4h) Ga4 (C2h) Ge4 (D2h) As4 (Td) Se4 (D2d)
S42+ (D4h)
Se42+ (D4h)
Calculated at the UB3LYP/631 + G(d) level [7]
a
Triphosphorus anion P3− (16e) was calculated to be linear (D∞h). [11]. Honea et al.
[12] prepared and isolated Si4 (16e) by lowenergy deposition into a solid nitrogen
matrix, and carried out a Raman spectra study to show that Si4 is a planar rhombus
(D2h). The Al44− tetraanion (16e) stabilized by the three Li+ ions in the most stable
structure of Li3Al4− is rectangular in a capped octahedral arrangement [13].
2.1.2 4N+ 2 Valence Electron Rule
Equilateral triangles (D3h) with 4N + 2 valence electrons in the singlet states are
the ground states of trimers [7].
The rule is applicable to all but S3 in Table 1 [7]. The most stable is thiozone
(C2v), whereas Se3 has D3h symmetry [14]. The lone pair repulsion may destabilize
the S3 ring and is weaker in the Se3 ring due to the smaller overlap between the
nonbonding orbitals.
Kuznetsov and Boldyrev [15] provided theoretical evidence that the B3−, Al3−,
and Ga3− anions (10e) have geometrical (cyclic, planar) and electronic (two delocalized p electrons) properties to be considered as aromatic systems. Positive cations
of all group XV trimers (14e), P3+ As3+ Sb3+ and Bi3+, have D3h equilateraltriangular
ground states [16].
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S. Inagaki
2.1.3 8N+ 2 Valence Electron Rule
8N+2 valence electrons do not allow square (D4h) as singlet ground states for
tetramers.
The 8N+ 2 rule has been completely substantiated by the calculated ground state
geometries of tetramers in Table 2 [7]. The Al42+ cluster (10e) is linear (D∞h)
[17]. The Si42− cluster (18e) has a butterfly structure (D2d) [18].
2.1.4 8N+ 6 Valence Electron Rule
Square structures with 8N+ 6 valence electrons are the ground states of
tetramers.
The 8N + 6 valence electron rule has been completely substantiated by the calculated fourmembered species in Table 2 [7]. Boldyrev, Wang, and their collaborators presented experimental and theoretical evidence of aromaticity in the Al42−[19]
Ga42− [20], In42− [20] and isoelectronic heterosystems, XAl3 [21]. The Al42− unit
(14e) was found to be square planar and to possess two p electrons, thus conforming to the (4n + 2) p electron counting rule for aromaticity. The p electron counting
rule would be more powerful if we could predict the number of p electrons of metal
atomic rings in an unequivocal manner. Our 8N + 6 electron rule only requires the
number of valence electrons in Al42−, which is easy to count.
Sundholm and coworkers [22] showed that (1) the squareshaped Al42− ring
sustains a very large diatropic ring current in an external magnetic field; (2) the
group XIII analogs, B42−, Ga42−, In42−, and Tl42− also exist and have D4h symmetry.
Fowler and coworkers [23] found that s electrons rather than p electrons contribute to the delocalized diamagnetic current in Al42− induced by a perpendicular
magnetic field shielding and concluded that Al42− is both s and paromatic. Zhan
et al. [24] theoretically emphasized the importance of the number of s electrons as
well as that of p electrons for unusual stability of Al3− and Al42−.
The Si42+ cluster (14e) was shown to be squareplanar (D4h) analogous to the
Al42− cluster [25].
The ground states of P42− [26, 27] and As42− [28] have D4h structures. Molecular
orbital analysis revealed that the square planar P42− dianion exhibits the characteristic
of paromaticity with six pelectrons [29]. The term lone pair aromaticity was
proposed for P42− [30]. Wang, Boldyrev, and their coworkers [26] presented
theoretical and experimental evidence for the squareplanar structures of Na+Pn42−(Pn
= P, As, Sb). The Sb42− [31] and Bi42− [32] dianions were prepared and shown to be
squareplanar (D4h).
The structures of the ground states of the S42+ [33, 34] and As42+ [35] dications
(22e) have D4h symmetry.
Although initially the aromaticity of Al42− was attributed to the two p electrons,
[19] it is now recognized that the contribution to aromaticity coming from the four
s electrons is more important than that from the p electrons [36–39].
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2.2 Alkali and Alkaline Earth Metals: 4N + 2 Valence Electron
Rule
The sorbital array of three and fourmembered rings is of the Hückel conjugation.
(Scheme 2). The splitting patterns of the orbital energy levels (Scheme 3) show that
the total number of valence electrons for the closedshell structures is 4N + 2 for
the three (N= 0) and fourmembered rings (N= 0, 1).
2.2.1 ThreeMembered Rings
The simplest metal cluster Li3 with two electrons (N= 0) is known to have a triangular
structure (D3h) as its global minimum, whereas Li3 of four electrons is linear [40].
The structure of the most stable, singlet states of Be3 with 4N + 2 valence electrons
(N= 1) is an equilateral triangle [41]. The Mg3 [42] clusters (6e) is a van der Waals complex of D3h symmetry. All of the three bonding and antibonding molecular orbitals are
occupied by a pair of electrons. No bonding nature can appear between the atoms. The
electronic structure is represented by three lone pairs in the sorbitals. This is the reason
the 4N+ 2 valence electron rule is applicable only for N= 0 in the case of threemembered rings. Mixingin of porbitals significantly contributes to the D3h structures.
2.2.2 FourMembered Rings
The Li42+ dication with two electrons (4N + 2, N= 0) adopts a tetrahedral structure
[42]. The single molecular orbital composed of four sorbitals at the lowest energy
level in the tetrahedron is lower than that in the square. The number of the inphase
relations between the sorbitals is greater in the tetrahedron.
The global minimum of a neutral Li4 molecule with 4N valence electrons (N= 1) does
not adopt a square structure (D4h) but a rhombus structure (D2h) [43]. The Raman spectroscopy supported the rhombic structure for the Li4 [44], Na4 and K4 clusters [45, 46].
The Mg42+ dication [42] with 4N + 2 (N= 1) valence electrons has a stable D4h structure
in agreement with the rule, but this is a local energy minimum. The linear structure is more
stable because it minimizes the Coulomb repulsion. This is in contrast to the tetrahedral
structure of the Li42 dication with two electrons (N= 0). The six electron systems cannot
form closedshell structures in the tetrahedron, but the two electron systems can do.
2.3 Larger Rings: Preference for Small Rings
p Bonds between heavy atoms are well known to be unstable relative to s bonds.
Large monocyclic rings tend to transform into polycylic structures by forming
stable s bonds between unstable p bonds.
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S. Inagaki
Density functional calculations showed transitions from planar to nonplanar
structures at n = 5 with increasing size of Aln and Gan clusters [47]. Both Si and
Al tend to build threedimensional structures rather than two or onedimensional
structures, except for n = 3 or 4 [48].
The planar cyclic P5− anion isoelectronic with cyclopentadienyl anion has been
prepared in the form of M+P5− salts (M = Li, Na) by Baudler et al. [49]. The pentaphosphole anion P5− favors planar D5h geometry [50] while the most stable structure
of P5+ is squarepyramidal [51]. The negatively charged pentamers Sb5 and Bi5 are
planar rings [52, 53].
2.4 Regular Octahedrons of M6 Clusters
There are Wade rules for metal clusters, [3, 4] which have been extended by Teo
[54, 55], Mingos [56, 57], and Jemmis [58, 59]. These general rules give only a
single number of electrons for a given polyhedron to be stabilized. The valence
electron rules for the three and fourmembered metal rings in the singlet states
(Sects. 2.1, 2.2) suggested that there could be more than one number of electrons. A valence electron rule was recently proposed for the regular octahedron of
highrow representative elements in the singlet states [8].
Atomic orbitals are separated into the sorbitals, the radial (r), and tangential (t)
porbitals (Scheme 2) [7]. The Hückel theory was applied to the sorbitals, the
radial (r), and tangential (t) porbitals of the regular octahedron. The qualitative
energy levels (Scheme 5) [8] show that the number of valence electrons is 6N+ 14
for the closedshell structures when all the sorbitals are occupied by two electrons.
The t1u prorbitals at the nonbonding level are allowed to interact with the bonding
ptorbitals of the same symmetry and are raised in energy above the nonbonding level. The upper limit of the number of electrons is 26 (N = 2).
The M6 clusters with 6N + 14 (N = 0–2) valence electrons assume regular octahedrons, whereas those with the other numbers of valence electrons do not.
The 6N + 14 (N= 0–2) valence electron rule was supported by the results of the
calculations of the M6 clusters of the third and fourth row elements at the
UB3LYP/631 + G(d) (Table 3) [8]. The regular octahedrons were located as
the energy minima for the 14 (N= 0) electron systems, Mg62− and Ca62−, for the 20
(N= 1) electron system, Al62−, and for the 26 (N= 2) electron systems, Si62− and
Ge62−. No energy minima were located for the regular octahedrons with 6N+ 14
(N≥ 3) or the other numbers of valence electrons than 6N + 14.
The 6N + 14 valence electron rule is based on the assumption that neighboring
prorbitals interact with each other more strongly than neighboring ptorbitals, or that
the a1g prorbital is lower in energy than the t2g ptorbitals (Scheme 5). When the interactions occur to a similar degree, the octahedral geometry of the 20 (N= 1) electron systems
is unstable. When the interaction between the ptorbitals is stronger, the regular octahedron prefers 18 and 20 (N= 0, 1) valence electron systems. The relative magnitudes of
the interactions between the pr and ptorbitals depend on the atoms.