Recoupled-Pair Bonding and 4-Electron 3-Center Bonding Units - The

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Recoupled-Pair Bonding and 4-Electron 3-Center Bonding Units Richard D. Harcourt* School of Chemistry, The University of Melbourne, Victoria 3010, Australia ABSTRACT: Consideration is given to recoupled-pair bonding and the origin of electronic hypervalence for formulations of the bonding for symmetric 4-electron 3-center ((4e,3c)) bonding units with one overlapping atomic orbital per atomic center. Molecular orbital and valence bond theory for symmetric (4e,3c) bonding units is redescribed and applied to aspects of the bonding for SF6 and CLi6. The results of minimal basis set calculations for CLi6 provide support for a hypothesis that two Li
C
Li (3e,3c) bonding units rather than two (4e,3c) bonding units are preferred for this molecule. Brief comments are also made on and as valence bond structures for the three electron bond. the use of

’ INTRODUCTION In ref 1, it is remarked that Glukhovtsev and Schleyer2 made the “sensible distinction” between geometric hypervalence or hypercoordination and electronic hypervalence. For a 4-electron 3-center ((4e,3c)) bonding unit with one overlapping atomic orbital (AO) per atomic center,3 the former does not necessarily require the latter. In ref 4, electron-pair recoupling has been used to account for hypervalence for sulfur, chlorine, and phosphorus atoms in the SFn (n = 1
6), ClFn (n = 1
7), and PFn (n = 1
5) series of compounds, without expansion of the sulfur, chlorine, or phosphorus valence shells (i.e., their 3d AOs do not participate as additional valence AOs). The two electrons that occupy a sulfur, chlorine, or phosphorus 3p AO, for example, which can overlap substantially with a 2p AO on each of two fluorine ligands, decouple to occupy separate right and left lobe orbitals and recouple with the unpaired electron on each of two fluorine atoms to participate in the formation of two (nonorthogonal) 2p
3p electron-pair bonds. In this paper, we shall show that, with one overlapping AO per atomic center, recoupled-pair bonding always occurs when an electron is delocalized from a double-occupied AO in a canonical Lewis VB structure (such as those of 1 and 3 in Figure 1) into either a 2-center bonding MO or a 2-center antibonding MO. A VB structure is thereby generated that involves increased-valence or electronic hypervalence for one of the atoms. Types of VB structures for octahedral SF6 and CLi6, without the participation in bonding of sulfur 3d orbitals as valence orbitals, are also discussed. These structures (structures 1 and 2 in each of Figures 2 and 3) involve two S
F or C
Li (2e,2c) electron-pair bonds, two (4e,3c) bonding units for SF6 and two (3e,3c) bonding units for CLi6. The results of VB calculations for CLi6 are reported to provide support for the hypothesis that two (3e,3c) rather than two (4e,3c) bonding units are preferred for this molecule. r 2011 American Chemical Society

’ ELECTRONIC HYPERVALENCE AND (4E,3C) BONDING a. Construction of Increased-Valence Structures. We redescribe and elaborate further on aspects of increased-valence theory for a symmetrical (4e,3c) bonding unit YAB, with Y equivalent to B and nuclear-centered overlapping AOs designated as y, a, and b. These (normalized) AOs can involve either minimal or extended basis sets, for example a = a0 and a = a0 þ ka00 þ..., and are oriented so that each of the AO overlap integrals is greater than zero. Initially, we shall use Heitler
London AO formulations of the wave functions for the Y
A, A
B, and Y
B electron-pair bonds in the canonical Lewis-type VB structures 1a,b, 3a,b and 4a,b of Figure 1. In the Lewis structures 4a and 4b, and also in structures 2 and 5 of Figure 1, the presence of a long or formal Y
B bond, , and is not displayed. In as in structures 2 and 5, this bond is fractional, (i.e., its bond-number is less than unity), as are the Y
A and A
B electron-pair bonds. Thin bond lines (—) are used to represent1,5
8 the latter two bonds. The VB structures 2 and 5 are obtained from the Lewis structures by means of the following one-electron delocalizations from a doubly occupied AO into either a bonding MO (ψba = b þ la or ψya = y þ la) or an antibonding MO (φ*ba = la
b or φ*ya = la
y), with 0 < l < ¥. 1a f 2: b f ψba

3a f 2: y f φba

4a f 2: a f ψba

3b f 5: y f ψya

1b f 5 : b f φya

4b f 5: a f ψya

Received: December 6, 2010 Revised: May 9, 2011 Published: May 26, 2011 6610

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Figure 1. Derivation of increased-valence structures for a (4e,3c) bonding unit. Atomic formal charges are not displayed in all YAB structures. In the text and in Figures 2
4, atomic formal charges for the non YAB structures are assigned on the assumption that bonding electrons are shared equally by pairs of adjacent atoms. With Heitler
London AO formulations of electron-pair bond wavefunctions, 2 t 1 T 4 and 5 t 3 T 4; cf. eqs 1 and 3 for symmetric YAB.

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The designation “increased-valence” has been associated with two effects.5
8 Using the previously published expressions for valence (refs 5a, 6, and 8), one can show that (i) The valence (VY, VA, or VB) for one of the atoms in VB structures 2 and 5 can exceed its value of unity in the types 1 and 3 Lewis structures. For example, for VB structure 2 with ψba = b þ la, from refs 5a, 6, and 8, we have VY = Vya þ Vyb = 1, VA = Vay þ Vab = 1/(l2 þ1) þ 2l2/(l2 þ1)2, and VB = Vba þ Vby = 2l2/(l2 þ1)2 þ l2/(l2 þ1). These formulas give VA = VB = 1 for l = 1, VA > 1 and VB < 1 for 0 < l < 1, and VA < 1 and VB > 1 for 1 < l < ¥. (ii) The number of electrons associated with Y
A, Y
B, and A
B bonding exceeds the value of 2, which occurs in each of the Lewis structures 1, 3, and 4. The maximum number of Y
A, Y
B, and A
B bonding electrons is 3, which occurs when l = 1 in either ψba = b þ la for VB structure 2 or ψya = y þ la for VB structure 5. b. Wave Functions for Increased-Valence Structures. For the 1a f 2, 4a f 2, 3b f 5, and 4b f 5 delocalizations, appropriate singlet (S = 0) spin wave functions for 2 and 5 are given9 by eqs 1 and 3. Ψ2 ¼ jy R a β ψba R b β j þ jaR y β bR ψba β j  Ψ1 þ lΨ4

ð1Þ


ðjy R ψba β ψba R ψba β j þ jψba R y β ψba R ψba β jÞ=ð1 þ llÞ ð2Þ Ψ5 ¼ jbR a β ψya R y β j þ jaR b β yR ψya β j  Ψ3 þ lΨ4

ð3Þ


ðjbR ψya β ψya R ψya β jþjψya R b β ψya R ψya β jÞ=ð1 þ llÞ ð4Þ

Figure 2. VB structures for octahedral SF6. The wave functions for S
F and C
Li electron-pair bonds (normal and fractional) in Figures 2
4 can be formulated using either Heitler
London AO or generalized Coulson
Fischer LMOs according to the nature of the discussion.

Figure 3. VB structures for octahedral CLi6. Each VB configuration for structures 2, 5, and 6 in Table 1 is equivalent to resonance between four equivalent increased-valence structures of these types.

VB structures 2 and 5 are examples of increased-valence structures1,5
8 for a (4e,3c) bonding unit.

With the antibonding MOs ψ*ba = l*b
a and ψ*ya = l*b
a orthogonal to their companion bonding MOs ψba = b þ la and ψya = y þ la (for which l* = (l þ x)/(1 þ lx) and x = Æy|aæ or Æb|aæ), the Ψ2 and Ψ5 of eqs 1 and 3 are equivalent to eqs 2 and 4, respectively.5a,b c. Recoupled-Pair Bonding in the Increased-Valence Structures. The (b)2 coupling in structure 1a is replaced by b
a (via ψba) and b
y couplings in 2. The y
a (via ψya) and y
b couplings in 5 replace the (y)2 coupling in structure 3b. The (a)2 coupling in structure 4a is replaced by a
b (via ψba) and a
y couplings in 2. The a
y (via ψya) and a
b couplings in 5 replace the (a)2 coupling in structure 4b. For 0 < l < ¥, the one-electron valencies Vab for 2 and Vay for 5, and the two-electron valencies Vay and Vby for 2 and Vab and Vby for 5, which arise from these diatomic couplings are, as indicated above, fractional. When the two A-atom electrons of the singlet diradical Lewis structure 4 are delocalized into separate ψya and ψba bonding MOs, we obtain the nonpaired spatial orbital10 (npso) VB structure 6 of Figure 1, with a
y (via ψya) and a
b couplings (via ψba) couplings. This VB structure can also exhibit electronic hypervalence for the central atom.11 Relationships between the wave functions for VB structures 2, 5, 6 and others, with separate generalized Coulson
Fischer type localized MOs (for example, ψab = a þ k0 b and ψba = b þ k00 a) used to formulate wave functions for Y
A and A
B electron-pair bonds, are provided in ref 12. In all cases, the responsibility for the electronic hypervalence of the A-atom arises from the contribution of the singlet diradical structure to the associated canonical Lewis structure resonance scheme. However, when the singlet diradical 6611

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structure is unimportant, geometric hypervalence, as distinct from electronic hypervalence, is still possible without the inclusion of the singlet diradical structure; see, for example, refs 13 and 14, for calculations of hydrogen bonding between two H2O molecules, and the stabilities of X3
with X = Cl, Br, and I.

’ (4E,3C) MO CONFIGURATION, SINGLET DIRADICAL, AND INCREASED-VALENCE STRUCTURES The relevance of singlet diradical structures for electronic hypervalence has not usually been considered explicitly in recent VB discussions of hypervalence, for example in refs 15
25. However, it is implied whenever reference is made to the MO formulation of (4e,3c) bonding, and in ref 15 because of the presence of (pseudo)11 one-electron bonds in some of the VB structures. Herndon26 has used increased-valence structures to describe the bonding for XeF2, XeF4, XeOF4, and XeF6, and therefore, singlet diradical structures contribute to the equivalent Lewis structure resonance scheme. We redescribe here aspects of the MO formulation of (4e,3c) bonding. Some of it has relevance for the VB descriptions of bonding for SF6 and CLi6, and for the discussions of hypervalent molecules presented in refs 27
32. With the canonical MOs of eq 5, ψ1 ¼ y þ k1 a þ b

ψ2 ¼ y
b

ψ3 ¼ y
k3 a þ b

ð5Þ it has been deduced that the (4e,3c) MO configuration (ψ1)2(ψ2)2 is equivalent to each of (a)
(d): (a) (ψya)2(ψba)2  (2y þ k1a)2(k1a þ 2b)2, which gives the VB structure 7 or 8,

with fractional electron-pair bonds. The fractionality arises because the Y
A and A
B localized MOs, ψya and ψba, are not orthogonal;33 there is an “overutilization”33 of the A-atom AO a. With AO overlap integrals omitted, the A-atom valence VA = Vay þ Vab for (ψ1)2(ψ2)2 is equal to34 16k12/{(k12 þ 4)(k12 þ 2)}, which has a maximum value of 1.3726 when k12 = 2 3 21/2. (b) Resonance between six canonical Lewis structures, according to eq 6

Therefore, the singlet diradical structure 4 of Figure 1 is a component of the canonical MO formulation of the wave function for (4e,3c) bonding, as was shown by Coulson.33 When the excited configurations (ψ1)1(ψ2)2(ψ3)1, (ψ2)2(ψ3)2, and (ψ1)2(ψ3)2 interact with (ψ1)2(ψ2)2, the lowest-energy MOCI wave function is equivalent12 to the lowest-energy linear combination of the wave functions for the six canonical Lewis structures of eq 6. (c) Resonance between increased-valence structures 2 and 5, or 9 and 1012

when (i)12 the localized MOs ψba = b þ 1/2k1a and ψya = y þ 1 /2k1a accommodate the electrons that form the one-electron

A
B and Y
A bonds of structures 9 and 10 and (ii)12 the localized MOs ψya = y þ 1/2k1a and ψay = a þ y/k1, and ψba = b þ 1/2k1a and ψab = a þ b/k1, replace the y, a, and b AOs that are used to accommodate the electrons of the fractional Y
A and A
B electron-pair bonds of 2 and 5. (d) Resonance between the npso structure 6 and the electron-pair bond VB structures 11 and 12,12

in which the Y
A and/or A
B bonding electrons occupy the localized MOs ψya = y þ k1a and ψba = b þ k1a. It has been suggested17 that (ψ1)2(ψ2)2 is equivalent to resonance between Lewis structures 1 and 3 with polar A
B and A
Y bonds, i.e., VB structures 11 and 12, with (in the most general case), separate localized MOs to accommodate each pair of bonding electrons. The singlet diradical structure 4 is thereby ignored. To conclude this section, we note that with orbitals ψya = y þ k00 a, ψay = a þ k0 y, ψba = b þ la, and b for increased-valence structure 9, and with orbitals y, ψya = y þ la, ψab = a þ k0 b, and ψba = b þ k00 a for increased-valence structure 10, the groundstate resonance between 9 and 10 (to give either a stable hypercoordinate species or a transition state) generates the lowest energy for the symmetric (4e,3c) bonding unit, when the parameters k0 , k00 , and l are chosen variationally.12

’ OCTAHEDRAL SF6 AND CLi6 a. Increased-Valence Structures. In ref 25a, the bonding for octahedral SF6 and CLi6 has been contrasted. It was concluded that (a) SF6 involves six localized, polar S
F bonds, as was suggested in ref 17, and (b) no evidence exists for the presence of six C
Li electron-pair bonds. The six polar S
F bonds of SF6 can be accommodated by resonance between VB structures that involve two (4e,3c) bonding units of type 8 above (i.e., of the type F;S;F with fractional (2e,2c) S
F bonds), and two polar, nonfractional (2e,2c) S
F bonds. In the spin-coupled VB studies of ref 27, six equivalent (extended basis set) sulfur orbitals are utilized for bonding in SF6. The VB structures of Figures 2 and 3 for SF6 and CLi6, as alternatives to the formulations of refs 25a and 27, will now be described. One can construct5a,35
37 Lewis structures with four (polar) S
F or C
Li electron-pair bonds, and either two F(
) ions or two Li atoms, as in Lewis structure 1 in each of Figures 2 and 3. These structures are associated with either two (4e,3c) or two (3e,3c) bonding units. In these VB structures, the AOs used by the sulfur or carbon atoms involve two sp hybrid AOs and two 3p or 2p AOs. From the Lewis octet structures of type 1 in each of Figures 2 and 3, with either AOs or Coulson
Fischer type localized MOs to accommodate the S
F or C
Li bonding electrons, increasedvalence structures of type 2 in these figures can be constructed via the one-electron delocalizations indicated in the type 1 structures.5a,35
37 Each of the type 2 VB structures possesses two (2e,2c) S
F or C
Li electron-pair bonds, and two (4e,3c) or (3e,3c) bonding units that involve one-electron bonds and fractional electron-pair bonds. Resonance between the 12 increased-valence structures of type 2 for each of SF6 and CLi6 can accommodate average distributions of atomic formal charges that range from S6þ þ (F
)6 and C4þ þ (Li
2/3)6 to S6
þ (Fþ)6 and C4- þ (Liþ2/3)6, according to the values of the localized MO polarity parameters. Thus the “ion cluster” or “metallic cage”25a 6612

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model for CLi6, with Liþq/6 for each lithium atom, C
q for carbon and q > 0, can be accommodated by resonance between the type 2 increased-valence structures of Figure 3. Inspection of the VB structures in Figures 2 and 3 shows that six (nonorthogonal) C
Li electron-pair bonds cannot be obtained for CLi6, which contains two fewer bonding electrons than has SF6. For CLi6, resonance between the 12 increased-valence structures of type 2 averages to produce five nonorthogonal, fractional electron-pair bonds and a one-electron bond. With Heitler
London AO wave functions for the long F
F bonds, Lewis structure 3 of Figure 2 involves a singlet diradical structure for each of the two Lewis structure resonance schemes, with finite values for at least some of the (4e,3c) MO polarity parameters. Lewis structure 4 of Figure 2 provides an alternative spin-pairing scheme for Lewis structure 3. The SF6 increased-valence structures of type 2 of Figure 2 can be stabilized further37 via the one-electron delocalizations indicated in 5, to give increased-valence structure 6 of Figure 2. However, because two fluorine atoms acquire þve formal charges, these delocalizations (which are also associated with (4e,3c) bonding units) should occur only to a small extent. Canonical structure formulations of (3e,3c) bonding units do not involve singlet diradical structures. For each Li
C
Li (3e,3c) bonding unit, with increased-valence structures such as and a Heitler
London AO wave function for the fractional C
Li electron-pair bond, the singlet diradical structure . The is replaced by the canonical structure36,38 canonical structure is also needed36,38 when Coulson
Fischer orbitals replace the Heitler
London AOs in . Increased-valence structures of type 3 in Figure 3, which are the ionic components of type 2 increased-valence structures, involve a (4e,3c) bonding unit and a (2e,3c) bonding unit. The (2e,3c) wave function (ψ1)2 is associated with the resonance. is used in 3. As an alternative to the two (3e,3c) bonding units associated with the CLi6 VB structures 1 and 2 of Figure 3, two (4e,3c) bonding units (as in SF6) arise in the Lewis and increased-valence structures 4 and 5 of Figure 3. The number of electron-pair bonds (fractional and normal) and the nature of the atomic formal charges (C(-) þ Li(þ1/2) þ Li(þ1/2) in 2, and Li(þ) þ Li(
1/2) þ Li(
1/2) in 5) suggest that the (3e,3c) bonding scheme of 2 is preferred to the (4e,3c) bonding scheme of 5. One other type of increased-valence structures requires consideration. There are 12 increased-valence structures of type 6 in Figure 3, each of which has one (3e,3c) and one (4e,3c) bonding unit. To complete the VB description for CLi6, we can write ΨðCLi6 Þ ¼ Ψ2 ð3Þ þ C3 Ψ3 ð6Þ þ C5 Ψ5 ð6Þ þ C6 Ψ6 ð12Þ ð7Þ in which the number of VB configurations (with 3-center MO formulations of the (2e,3c), (3e,3c), and (4e,3c) bonding units) is given in parentheses with each Ψ. The results of the VB calculations reported in the next section provide support for the hypothesis that the primary VB structures for the ground state are of type 2. b. STO-6G Calculations for CLi6 with (3e,3c) and (4e,3c) Bonding Units. The VB methodology described in Appendix 1 has been used to calculate the CLi6 energies and bond parameters reported in Table 1.

Table 1. CLi6 Bond Parameters (i) j in σ(CLi) = (sp)C þ j(2s)Li, (ii) k1 and k3 in ψ1 = y þ k1a þ b and ψ3 = y
k3a þ b with y = (2s)Li(Y), a = (2pσ)C(A), b =
(2s)Li(B), k3 = 2(1
Æy|bæ þ k1Æy|aæ)/(k1 þ 2Æy|aæ), and Energies (E, au) type of VB structure 2

a

no. of VB configurations

k

k1

k3

E

3

0.5

0.7

1.55761


82.05824a

3

6

0.4

0.7

1.55761


82.04420

5

6

0.3

2.1

0.84266


81.38841

5

30

0.3

2.1

0.84266


81.43734

5

54

0.3

2.1

0.84266


81.46057

6 2T3

12 9

0.1 0.5

1.2 0.7

1.15120 1.55761


81.93499
82.05826

This value replaces the
82.0500 au reported in ref 36.

With or without configuration interaction (CI), the energy of the (4e,3c) formulation associated with resonance between increased-valence structures of type 5 is substantially higher than that of the (3e,3c) formulation, for resonance between increasedvalence structures of type 2. Without CI, the energy for resonance between increased-valence structures of type 6 is intermediate between those for the 2 and 5 resonances. The 2 T 3 covalent-ionic resonance generates only a slight stabilization of the type 2 increased-valence structures. See Appendix 2 for the results of calculations for other types of CLi6 VB structures.

’ OTHER COMMENTS ON (4E,3C) BONDING UNITS AND SINGLET DIRADICAL STRUCTURES Braida and Hiberty,14 who have included the singlet diradical structure 4 of Figure 1 in their VB calculations for X3
anions with X = H or halogen, have remarked that the logic of the Rundle
Pimentel3 model fails in the case of H3
, which is unstable although the MOs ψ1 and ψ2 are bonding and nonbonding. Electronic hypervalence, which is associated34 with the (ψ1)2(ψ2)2 formulation3 of the (4e,3c) bonding unit, does not guarantee stable geometric hypervalence, in the same way that geometric hypervalence or hypercoordination does not necessarily require electronic hypervalence.13,14 Without reference to increased-valence structures, singlet diradical structures are included in the VB descriptions of (4e,3c) bonding units considered in ref 39. is used to In Scheme 6 of ref 40, the VB structure represent the three-electron bond for the π-electron system of the allyl anion, which has been calculated to have appreciable is also used in singlet diradical character (see also ref 41). ref 39. The top dot in this structure represents the antibonding provides an alternative to the Green
Linnett42 electron. VB structure . The latter structure involves two electrons with parallel spins (net antibonding when AO overlap is included)43 and one bonding electron with an opposed spin, as with the  and o representing electrons with þ1/2 and in
1/2 spin quantum numbers. The Green
Linnett structure forms the basis for the construction of the (4e,3c) increased-valence structures (cf. eqs 1
4 above for the wave function identities for (4e,3c) bonding units.) The use of to provide the VB structure or for a (4e,3c) bonding unit is discussed in ref 44. By replacing with , increased-valence structure 2 of Figure 1 is obtained 6613

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The Journal of Physical Chemistry A via the electron spin-pairings. Using the latter structure (with the long Y
B bond omitted), and structure 5 of Figure 1, we obtain the increased-valence structures and for the allyl anion. Each of ClF3, ClF4, ClF5, SF4, SF5, and SF6, without 3d AO participation, involves at least one (4e,3c) bonding unit (cf. ref 5a, p 213 for the increased-valence structures). In refs 1a, 1b, 6, 7, and 33 the stabilities of PF5, ClF3, and XeF2 and the nonexistence of NF5, FF3, and NeF2 are related to the smaller ionization potentials for A = P, Cl, and Xe relative to A = N, F and Ne in the singlet diradical structure 4. The singlet diradical structure is dominant when the A atom ionization potential is large, and an effective (4e,3c) bonding units is not then established. The results of STO-3G VB calculations45 for PF5 and NF5, with two sets of geometries for each molecule, give a substantially than for . It is noted that larger weight for D3h symmetry NF5 has been calculated to correspond to a local minimum.46 See also refs 47 and 48 for recent VB studies of (4e,3c) bonding units in 1,3-dipolar (or “zwitterionic diradical hybrid”5a) type molecules.

’ CONCLUSIONS When a Lewis VB octet structure for either a hypercoordinate (for example, SF6 and CLi6) or a nonhypercoordinate molecule (for example, N2O or F2O2)5
8,35b,44a involves at least one (4e,3c) or (3e,3c) bonding unit, the associated increased-valence structure always involves an apparent violation of the Lewis
Langmuir octet rule for at least one of the atoms.1a,c,5
7 For a molecule with a (4e,3c) bonding unit, the YAB increased-valence theory described above shows that this effect is a consequence of including the singlet diradical structure as well as the standard Lewis octet structures (such as structure 1 for SF6 in Figure 2) in the component Lewis structure resonance scheme. Finally, it is to be noted that when the AO a is replaced by the AOs a0 and a00 , there are 15 S = 0 spin VB structures. With ) used twice, 11 of these structures form (yy)(bb) (i.e., two sets of VB structures for (4e,3c) bonding, i.e., those of eq 8,

which use either a0 or a00 . The remaining four structures, (ya0 )(a00 b), (yb)(a0 a00 ), (yy)(a0 a00 ), and (bb)(a0 a00 ), with both a0 and a00 singly occupied, contribute to the 15 structure resonance scheme, which involves nine variational parameters. As well as the electronic hypervalence that is associated with each of the two (4e,3c) bonding units, it can occur also in the (ya0 )(a00 b) structure with the y and b electrons S = 0 spin-paired with the a0 and a00 electrons, respectively.49 And if two Ψ(4e,3c) wave functions are constructed using the AOs y, a1 = a0 þ ka00 , b and y, a2 = k*a0
a00 , b (with a2 orthogonal to a1) and combined linearly, a linear combination of the wave functions for the 15 VB structures is obtained, with five variational parameters.

’ APPENDIX 1 Methodology for Calculations for Octahedral CLi6. To provide provisional support for the preference for use of the CLi6 (3e,3c) VB structures of type 2 over the (4e,3c) VB

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structures of type 5 in Figure 3, Roso’s ab initio VB program50 was used to perform elementary STO-6G calculations, with C
Li bond lengths of 2.024 Å,51 and “best-atom” exponents.52 The lithium valence AOs were assumed to be 2s AOs, i.e., with no 2s
2p hybridization. For each of the (4e,3c) and (3e,3c) calculations, use is made of the (4e,3c) and (3e,3c) MO configurations (ψ1)2(ψ2)2 and (ψ1)2(ψ2)1, which provide12,36 a restricted form of resonance between increased-valence structures (cf. eq 6 and VB structures 9 and 10 for (4e,3c) bonding units). The values for the parameter k1 in the ψI of eq 5 and the k of σ(CLi) = (sp)C þ k(2s)Li were determined variationally. The VB calculation for resonance between increased-valence structures of type 2 in Figure 3, each with two (3e,3c) bonding units, involves the formation of a linear combination of three degenerate VB configurations. Each of them possesses one of the S = 0 spin (3e,3c) MO bonding units (ψ1)x2(ψ2)x1(ψ1)y2(ψ2)y1, (ψ1)x2(ψ2)x1(ψ1)z2(ψ2)z1 and (ψ1)y2(ψ2)y1(ψ1)z2(ψ2)z1, together with double-occupation of two σ(CLi) orbitals, as well as the 1s AOs. For the (4e,3c) bonding units associated with the increasedvalence structures of type 5 in Figure 3, the MO configurations are (ψ1)x2(ψ2)x2(ψ1)y2(ψ2)y2, (ψ1)x2(ψ2)x2(ψ1)z2(ψ2)z2, and (ψ1)y2(ψ2)y2(ψ1)z2(ψ2)z2, each of which also involves double occupation of one σ(CLi) orbital and the 1s AOs, to give six VB configurations. Two sets of CI calculations were performed. They included: (a) The 12 (S = 0 spin) singly excited configurations that arise from (ψ1)x f (ψ3)x, (ψ1)y f (ψ3)y, and (ψ1)z f (ψ3)z excitations, and the 12 closed-shell doubly excited MO configurations that arise from HOMO f LUMO excitations. (b) The excited configurations of (a), together with the remaining doubly excited configurations. Both Rumer S = 0 spin pairing schemes have been included for the doubly excited configurations that involve four singlyoccupied MOs. For each of the (4e,3c) calculations, the groundstate wavefunction is calculated to be triply-degenerate. Twelve degenerate VB configurations, each with two singlyoccupied orbitals (a ψ2-type MO and a σ(CLi) MO), were used to calculate the energy for resonance between increased-valence structures of type 6 of Figure 3, with one (3e,3c) and one (4e,3c) bonding unit. For these configurations, there are two types of ψ1 type 3-center MOs—one for each of the (4e,3c) and (3e,3c) bonding units—and two types of σ(CLi) MOs according to whether a σ(CLi) is singly or doubly occupied. Therefore, there are four variational parameters: k10 , k20 , k10 , and k20 . For Table 1, it has been assumed that k10 = k20 = k1 and k10 = k20 = k. Similarly, for increased-valence structures of type 3 of Figure 3, there are four variational parameters, which have been reduced to two for Table 1.

’ APPENDIX 2 Additional VB Calculations for CLi6. (a) Interaction between the 24 (4e,3c) excited singlet configurations of the types {(ψ1)x}2{(ψ2)x}1{(ψ3)x}1{(ψ1)y}2{(ψ2)y}1{(ψ3)y}1 and {(ψ3)x}2{(ψ2)x}1{(ψ1)x}1{(ψ3)y}2{(ψ2)y}1{(ψ1)y}1 gives an energy of
81.460 73 au (with triple degeneracy), which is lower than the
81.460 57 au of Table 1. (Both energies lie well above the
82.058 24 au of that Table.) Unitary transformations of the MOs in S = 0 spin MO wavefunctions such as |ψ1Rψ1βψ2Rψ3β| 6614

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Figure 4. Types of VB structures for octahedral CLi6, with all electrons accommodated in separate nonorthogonal C
Li 2-center MOs. Each of the four electron-pair bonds is a fractional electron-pair bond.

þ |ψ1Rψ1βψ3Rψ2β| show that these wavefunctions are associated with the

ARTICLE

(8) Harcourt, R. D. J. Am. Chem. Soc. 1978, 100, 8060. 1979, 101, 5456. In this reference and in ref 5a, p 166, ψba = kb þ a and AO overlap integrals are omitted. They have been included in ref 1a. In eqs 1
4 above, and ref 9 below, l is equal to 1/k. (9) With the diatomic (φ*ba)1(a)1(b)1 and (φ*ya)1(a)1(y)1 configurations associated with the 3a f 2, and 1b f 5 delocalizations, the S = 0 spin wave functions jyR φba β aR bβ j þ jφba R yβ bR aβ j þ jy R φba β bR aβ j þ jφba R y β aR b β j ¼ jyR aβ ψba R bβ j þ jaR y β bR ψba β j  Ψ1 þ lΨ4 and jbR φya β aR yβ j þ jφya R bβ y R aβ j þ jbR φya β y R aβ j þ jφya R b β aR y β j ¼ jbR aβ ψya R y β j þ jaR bβ yR ψya β j  Ψ3 þ lΨ4

resonance scheme, rather than that required for resonance between increased-valence structures. (b) When six non-orthogonal C
Li orbitals of the type σ(CLi) = ψCLi = 2sC þ λ2pC þ k2sLi are used to accommodate the valence electrons of VB structures 1 and 2 of Figure 4, there are 12 VB structures of type 1 and three VB structures of type 2. With λ = 0.6 and k = 0.8, a minimum energy of
81.841 80 au is obtained, which lies above the energy of
82.058 24 au for resonance between the three increased-valence structures of type 2 in Figure 3. The Coulson
Chirgwin weights for structures 1 and 2 of Figure 4 are 0.064 35 and 0.075 94, respectively.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT I am indebted to and thank Dr. Walter Roso for providing me with his ab initio VB program. ’ REFERENCES (1) (a) Harcourt, R. D. J. Mol. Struct. 1993, 300, 245. (b) Schulz, A. Trends. Inorg. Chem. 1999, 6, 137. (c) Seok, W. K.; Klap€otke, T. M. Bull. Korean Chem. Soc. 2010, 31, 781. (2) Glukhovtsev, M. N.; Schleyer, P. v. R. Chem. Phys. Lett. 1992, 198, 547. See also(a) Schleyer, P. v. R. Chem. Eng. News 1984, 62, 4. (b) Harcourt, R. D. Chem. Eng. News 1985, 63, 3. (3) (a) Pimentel, G. C. J. Chem. Phys. 1951, 19, 446. (b) Hach, R. J.; Rundle, R. E. J. Am. Chem. Soc. 1951, 73, 4321. (c) Musher, G. I. Angew. Chem., Int. Ed. 1969, 8, 54. (4) (a) Woon, D. E.; Dunning, T. H., Jr. J. Phys. Chem. A 2009, 113, 7915. (b) Chen, L.; Woon, D. E.; Dunning, T. H., Jr. J. Phys. Chem. A 2009, 113, 12645. (c) Woon, D. E.; Dunning, T. H., Jr. J. Phys. Chem. A 2010, 114, 8845. (5) (a) Harcourt, R. D. The Electronic Structures of Electron-Rich Molecules; The Pauling “3-Electron Bond” and “Increased-valence” Theory; Springer-Verlag: Heidelberg, 1982; Vol. 30. A 2003 update is available from the author. (b) Harcourt, R. D. In Quantum Mechanical Methods in Main-Group Chemistry; Klap€otke, T. M., Schulz, A., Eds.; Wiley: Chichester, U.K., 1998; p 226. (c) Klap€otke, T. M. (in Moderne Anorganische Chemie, 3rd ed.; Riedel, E., Ed.; de Gruyter: Berlin, 2007) has provided theory for and examples of increased-valence structures. See also ref 1b above. (6) Harcourt, R. D. Int. J. Quantum Chem. 1996, 60, 553. In the Introduction, a discussion is provided for several pre-1996 approaches to electronic hypervalence. (7) Harcourt, R. D. Eur. J. Inorg. Chem. 2000, 1901.

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