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1 Three- and Four-Membered Atomic Rings

1 Three- and Four-Membered Atomic Rings

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Orbitals in Inorganic Chemistry



295



D 3h



D 4h



Oh



Scheme 1  Symmetries of metal clusters for (r, t, p) approach



Hückel



Hückel



Möbius



Hückel



s



pr



pt







Hückel



Hückel



Hückel



Hückel



s















s











Scheme 2  Hückel or Möbius conjugation of the arrays of s-orbitals and radial (pr), tangential

(pt), and perpendicular (pp) p-orbitals



(Scheme 3). The qualitative energy levels (Scheme 4) show the number of valence

electrons necessary to obtain closed-shell electronic structures. Each orbital in

the s-orbital set is assumed to be occupied by a pair of electrons since the s-orbital

energies are low and separate from those of the p-orbital ones, especially for

heavy atoms. The total number of valence electrons for the closed-shell structures



296



S. Inagaki



a



b



c



Hückel



Möbius



Hückel



Scheme 3  Energy splittings of three- and four-orbital 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















Three-membered ring



s















Four-membered ring



Scheme 4  Orbital energy levels



is 4N+ 2 for the three-membered rings and 8N+ 6 for the four-membered 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 four-membered

ring structures of D3h and D4h symmetry, respectively.



Orbitals in Inorganic Chemistry



297



Table 1  Ground state geometries of trimersa

Number of valence electrons

4N+ 2

10



14



Al 3 (D3h)

Ga-3 (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/6-31 + 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/6-31 + 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 low-energy 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 equilateral-triangular

ground states [16].



298



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 four-membered 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 co-workers [22] showed that (1) the square-shaped 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 co-workers [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 p-aromatic. 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 square-planar (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 p-aromaticity with six p-electrons [29]. The term lone pair aromaticity was

proposed for P42− [30]. Wang, Boldyrev, and their co-workers [26] presented

theoretical and experimental evidence for the square-planar structures of Na+Pn42−(Pn

= P, As, Sb). The Sb42− [31] and Bi42− [32] dianions were prepared and shown to be

square-planar (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].



Orbitals in Inorganic Chemistry



299



2.2 Alkali and Alkaline Earth Metals: 4N + 2 Valence Electron

Rule

The s-orbital array of three and four-membered 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 closed-shell structures is 4N + 2 for

the three- (N= 0) and four-membered rings (N= 0, 1).

2.2.1  Three-Membered 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 s-orbitals. This is the reason

the 4N+ 2 valence electron rule is applicable only for N= 0 in the case of three-membered rings. Mixing-in of p-orbitals significantly contributes to the D3h structures.

2.2.2  Four-Membered Rings

The Li42+ dication with two electrons (4N + 2, N= 0) adopts a tetrahedral structure

[42]. The single molecular orbital composed of four s-orbitals at the lowest energy

level in the tetrahedron is lower than that in the square. The number of the in-phase

relations between the s-orbitals 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 closed-shell 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.



300



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 three-dimensional structures rather than two- or one-dimensional

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 square-pyramidal [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 four-membered 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

high-row representative elements in the singlet states [8].

Atomic orbitals are separated into the s-orbitals, the radial (r), and tangential (t)

p-orbitals (Scheme 2) [7]. The Hückel theory was applied to the s-orbitals, the

radial (r), and tangential (t) p-orbitals of the regular octahedron. The qualitative

energy levels (Scheme 5) [8] show that the number of valence electrons is 6N+ 14

for the closed-shell structures when all the s-orbitals are occupied by two electrons.

The t1u pr-orbitals at the nonbonding level are allowed to interact with the bonding

pt-orbitals 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/6-31 + 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

pr-orbitals interact with each other more strongly than neighboring pt-orbitals, or that

the a1g pr-orbital is lower in energy than the t2g pt-orbitals (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 pt-orbitals 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 pt-orbitals depend on the atoms.



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