J. Phys. Chem. A 2010, 114, 8573–8580
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Valence Bond Structures for Three-Electron Three-Center and Four-Electron Three-Center Bonding Units: Some Further Examples† Richard D. Harcourt* School of Chemistry, The UniVersity of Melbourne, Victoria 3010, Australia ReceiVed: NoVember 28, 2009
A formula is derived for the R-atom valence in the valence bond structure X · R:Y, with one overlapping atomic orbital per atomic center. Using this formula, the results of STO-6G valence bond calculations for linear H3, HFH, FHF, and F3 transition states show that for each molecule, the R-atom exhibits increased˙ R:Y, with no valence or electronic hypervalence, that is, its valence exceeds that for the Lewis structure X one-electron bond. Consideration is also given to (a) valence bond formulations for the conversion of the excited states of the reactants into the excited states of the products for the X• + R:Y f X:R + Y• and X:(-) + R:Y f X:R + Y:(-) reactions and the state correlation diagram for X• + R:Y f X:R + Y•, (b) Kekule´ and Dewar increased-valence structures for D6h symmetry systems, and (c) the three-center molecular orbital interaction of the X• electron with the antibonding molecular orbital of R:Y, to form the one-electron X-R bond of X · R:Y. Introduction In refs 1–3, the valence bond (VB) formulation of Scheme 1 has been provided for radical exchange reactions of the general type X• + R:Y f X:R + Y•. As discussed in more detail in refs 1–3, a one-electron transfer in this formulation leads to the formation of either a reactantlike complex (RC) with VB structure II or a product-like complex (PC) with VB structure III or to the decomposition of the latter to form the products of IV. For finite internuclear separations, the (lower-energy) wave function for resonance between VB structures II and III, Ψ ) ΦRC + FΦPC, provides1–3 a variationally best energy for the XRY entity, either as a transition state for the X• + R:Y f X:R + Y• reaction or as a stable radical. Relative to the Lewis VB structures I and IV, VB structures II and III are examples of three-electron three-center increased-valence structures.1–3 In the next section, wave functions for VB structures II and III will be presented (cf. refs 1–3). They are needed subsequently for consideration of the following topics on threeelectron three-center bonding. Each topic can be read independently of the others. (a) The derivation of an equation for the R-atom valence for VB structure II, with all atomic orbital (AO) overlap integrals included. Using this equation, STO-6G estimates of V(r) will then be reported for linear H3, HFH, FHF, and F3 transition states. (b) VB formulations for the conversion of the excited states of the reactants into the excited states of the products for the X• + R:Y f X:R + Y• and X:(-) + R:Y f X:R + Y:(-) reactions and the construction of the state correlation diagram for the X• + R:Y f X:R + Y• reaction. (c) Kekule´ and Dewar increased-valence structures for D6h systems to demonstrate that the energy for resonance between six Dewar increased-valence structures can lie below the energy for resonance between six Kekule´ increased-valence structures. †
Part of the “Klaus Ruedenberg Festschrift”. * To whom correspondence should be addressed. E-mail: r.harcourt@ unimelb.edu.au.
SCHEME 1: VB Formulation for the Reaction X• + R:Y f X:R + Y•
Each of the increased-valence structures involves two threeelectron three-center bonding units. (d) Three-center MO interaction of the X• electron orbital with the antibonding molecular orbital (MO) of R:Y. Although this paper is concerned primarily with aspects of three-electron three-center bonding (see, for example, ref 4 for alternative VB formulations), some discussion of four-electron three-center bonding will also be presented in (b). For any linear combination ciΨi + cjΨj of normalized wave functions Ψi and Ψj, weights Wi and Wj will be calculated using the formulas5 Wi ) ci2/(ci2 + cj2) and Wj ) cj2/(ci2 + cj2). For a one-electron bond, with i ) a and j ) b, these formulas correspond to the A- and B-atom AO charges Paa and Pbb. Wave Functions for the Reactant-Like and Product-Like Complexes II and III In the simplest treatment of the three-electron three-center bonding for Scheme 1, each of X, R, and Y contributes one (normalized, hybrid or nonhybrid) overlapping atomic orbital (AO) s designated as x, r and y s to the bonding scheme. These AOs are oriented so that the overlap integrals Sxr ) 〈x|r〉 and Sry ) 〈r|y〉 are both greater than 0 for II and III. The electrons occupy the (nonorthogonal) MOs φxr ) x + lr, φ′ry ) r + k′y, φ′′yr ) y + k′′r in VB structure II and φ′rx ) r + κ′x, φ′′xr ) x + κ′′r, and φyr ) y + λr in VB structure III. For a symmetrical transition state, λ ) l, κ′ ) k′, and κ′′ ) k′′. The S ) MS ) 1/2 spin wave function for the reactant-like complex II can be expressed1c,2,3 according to eq 1
10.1021/jp911294x 2010 American Chemical Society Published on Web 02/05/2010
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ΦRC ) Φ′RC + µΦ′′RC
(1)
in which
Φ′RC ) φxrRφ′ryRφ′′yrβ + φxrRφ′′yrRφ′ryβ
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(2)
Φ′′RC ) 2 φxrβφ′ryRφ′′yrR - φxrRφ′ryβφ′′yrR -
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φxrRφ′ryRφ′′yrβ
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(3)
With eight canonical Lewis structures (Figure 1), there are seven variational parameters (l, k′, k′′, λ, κ′, κ′′ and F) for Ψ ) ΦRC + FΦPC at intermediate stages along the reaction coordinate. Therefore, the parameter µ is not required to be a variational parameter. It can be chosen1c,3 so that ΦRC ) Φ′RC + µΦ′′RC is orthogonal to the wave function ΦPC ) Φ′PC + νΦ′′PC (with ν ) µ) for the product-like complex III. However, here, we shall assign a value of 0 to µ, to give ΦRC ) Φ′RC. For this wave function (eq 2), the two electrons of the fractional R:Y electron pair bond have opposed spins. Using eq 2, and a corresponding wave function for the ΦPC (eq 4, in which the X-R electrons are spin-paired) ΦPC ) Φ′PC ) φyrRφ′rxRφ′′xrβ + φyrRφ′′xrRφ′rxβ
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(4)
we can formulate the reaction mechanism of Scheme 1 according to Scheme 2, in which crosses and circles (× and O) represent electrons with ms ) +1/2 and -1/2 spin quantum numbers. The ΦRC and ΦPC of eqs 2 and 4 can be expressed according to eqs 5 and 6
ΦRC ) (1 + k'k'')Φ1 + 2k''Φ2 + 2k''Φ3 + l(1 + k'k'')Φ5 + 2k'lΦ7
(5)
ΦPC ) (1 + κ′κ′′)Φ4 - 2κ′′Φ5 + 2κ′Φ6 λ(1 + κ′κ′′)Φ2 + 2κ′λΦ8
(6)
that is, as linear combinations of the wave functions for five of the eight canonical structures of Figure 1. For each of the H3, HFH, FHF, and F3 transition states, the calculated value of k′k′′ (Table 2) is substantially smaller than unity. Therefore, to simplify the subsequent treatment, this term will be omitted from the ΦRC of eq 5.
Figure 1. Eight canonical XRY Lewis structures3 with S ) MS ) 1/2 spin wave functions for the three active-space AOs x, r, and y (cf. refs 1–3). C ) doubly occupied AOs for the core electrons. In refs 1–3, Φ5 is defined as Φ5 ) |CyRrRrβ| and +l(1 + k′k′′)Φ5 in eq 5 is replaced by -l(1 + k′k′′)Φ5. Therefore C2 ) +C5 in Table 2.
SCHEME 2: Scheme 1 with Electron Spins Included for µ ) ν ) 0 in ΦRC ) Φ′RC + µΦ′′RC and ΦPC ) Φ′PC + νΦ′′PCa
a When µ * 0 and ν * 0, parallel spins as well as opposed spins also contribute to the R:Y and X:R.
definitions of atomic valence are provided, for example, in ref 8 and references therein. When the electron pair bond wave function is expressed according to eq 9
Ψ(A:B) ) c1{a(1)b(2) + b(1)a(2)} + c2a(1)a(2) + c3b(1)b(2) (9) the (Wiberg)6 valence of eq 7 can be expressed9 according to eqs 10 and 11
Vab ) PaaPbb ) 2c12 /D + 2c12c22 /D2 + 2c12c32 /D2 + 4c22c32 /D2 (10) ) Vab(cov)11 + Vab(cov,ion)12 + Vab(cov,ion)13 + Vab(ion,ion)23 (11)
Vab ) PaaPbb ) Paa(2 - Paa)
(7)
Vab ) 2PaaPbb ) 2Paa(1 - Paa)
(8)
in which D ) 2c12 + c22 + c32. When both electrons are bonding and the AO overlap integral Sab is greater than 0, the coefficients c1, c2, and c3 of eq 9 have the same signs. However, if two of the products c1c2, c1c3, or c2c3 have negative signs, then the associated (antibonding) covalent-ionic and/or ionic-ionic resonance term in eq 10 is assigned a negative sign.9 This approach is needed for H3, for which an STO-6G estimate of k′′ in the MO φ′′yr ) y + k′′r is negative (cf. Table 2 below). (b) Valence V(ry) for Structure II in Scheme 2. To evaluate the valence V(ry), we express the ΦRC of eq 5 (with k′k′′ omitted) according to eq 12
(cf. refs 6 and 7) will be used to calculate atomic valencies, Vab. The Paa and Pbb are the (normalized) a and b AO charges for either A:B (eq 7) or A · B (eq 8). Other approaches to
ΦRC ) (Φ1 + 2k''Φ2 + 2k'Φ3) + l(Φ5 + 2k'Φ7) ) (1/NI)ΦI(n) + (l/NV)ΦV(n) (12)
R-Atom Valence for the Reactant-Like Complex II of Scheme 2 (a) Formulas for Atomic Valence. For diatomic electron pair and one-electron bonds, A:B and A · B, eqs 7 and 8
Further Examples of Valence Bond Structures
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ΦRC ) Φ1′ + 2k''Φ2 + (Φ1′′ + lΦ5) + 2k'(Φ3 + lΦ7) ) Φ1′ + 2k''Φ2 + ΦVI + 2k'ΦVII ) (1/N1′)Φ1′(n) + (2k''/N2)Φ2(n) + (1/NVI)ΦVI(n) + (2k'/NVII)ΦVII(n)
(21)
in which Φ1′ ) |CxRrRyβ|, Φ1′′ ) |CxRyRrβ|, 1/NVI2 ) S1′′1′′ + 2lS1′′5 + l2S55, and 1/NVII2 ) S33 + 2lS37 + l2S77. The resulting expression for V(rx) is given by eq 22 Figure 2. One-electron bond VB identities. ΦV ) Φ5 + 2k′Φ7 ) |CrRyR(y + 2k′r)β| ∝ |C{2(k′*)y - r}R(y + 2k′r)R(y + 2k′r)β|.
(13) (14)
VVI(rx) ) VVII(rx) ) 2l2 /(l2 + 1)2
1/NI2 ) S11 + 4k''S12 + 4k'S13 + 4k''2S22 + 8k''k'S23 +
1/NV2 ) S55 + 4k'S57 + 4k'2S77
with Sij ) 〈Φi|Φj〉. The ΦI(n) and ΦV(n) are normalized wave functions for VB structures I and V of Scheme 2 and Figure 2, respectively. The resulting expression for V(ry) is given by eq 15
V(ry) ) WIVI(ry) + WVVV(ry)
(15)
in which WI ) (1/NI)2/Dry, WV ) (l/NV)2/Dry, and Dry ) (1/NI)2 + (l/NV)2. Using eqs 7 and 8 to calculate AO charges when k′ and k′′ are both greater than 0, we obtain the AO charges of eqs 16a and 16b for ΦI(n) and eq 17 for ΦV(n). The resulting expressions for VI(ry) and VV(ry) are then given by eqs 18 and 19.
Prr ) (1 + 4k''2)/(1 + 2k''2 + 2k'2)
(16a)
Pyy ) (1 + 4k'2)/(1 + 2k''2 + 2k'2)
(16b)
Prr ) 1/(1 + 4k'2)
Pyy ) 4k'2 /(1 + 4k'2)
(17) VI(ry) ) (1 + 4k''2)(1 + 4k'2)/(1 + 2k''2 + 2k'2)2
(18) VV(ry) ) 8k'2 /(1 + 4k'2)2
(19)
For H3, k′′ is less than 0 (Table 2). Therefore -2c1 c2 /D and -4c22c32/D2 replace +2c12c22/D2 and +4c22c32/D2 in eq 10. The VI(ry) is then given by eq 20. 2
(22)
in which WVI ) (1/NVI)2/Drx, WVII ) (1/NVII)2/Drx, and Drx ) 1/N1′2 + (2k′′/N2)2 + 1/NVI2 + (2k′/NVII)2. Each of the VB structures VI and VII of Figure 2 has a oneelectron R-X bond (Figure 2). The valencies for these structures are given by eq 23.
in which
4k'2S33
V(rx) ) WVIVVI(rx) + WVIIVVII(rx)
2
2
VI(ry) ) {1 + 4k'2(1 - 4k''2)}/(1 + 2k''2 + 2k'2)2
(20) (c) V(rx) for VB Structure II in Scheme 2. To determine the valence V(rx), we express the ΦRC as
(23)
(d) STO-6G Calculations of V(r) for H3, HFH, FHF, and F3. Using eqs 5 and 6 for ΦRC and ΦPC, with κ′ ) k′, κ′′ ) k′′, and λ ) l for a symmetric transition state, gives
ΨTS(II T III) ) ΦRC - ΦPC ) (1 + k'k'')(Φ1 - Φ4) + {2k'' + l(1 + k'k'')}(Φ2 + Φ5) + 2k'(Φ3 - Φ6) + 2k'l(Φ7 - Φ8) (24) ≡ C1(Φ1 - Φ4) + C2(Φ2 + Φ5) + C3(Φ3 - Φ6) + C4(Φ7 - Φ8)
(25)
With k′k′′ set equal to 0 in eq 24, the k′, k′′, and l parameters are calculated from the Ci of eq 25. The STO-6G procedure used in ref 3 to calculate ΨTS(II T III) for HFH and FHF is also used to calculate the corresponding wave functions and parameters for the F3 and H3. The four wave functions are reported in Table 1. In Table 2, the V(ry) and V(rx) are reported, together with the parameters that are needed to calculate them. The results of the calculations show that as l increases from 0, to form the one-electron X-R bond, V(rx) and V(ry) respectively increase and decrease, and V(r) exceeds V(ry)(l ) 0) to give increased-valence or electronic hypervalence for the R-atom. Except for H3, the calculated valence for this atom exceeds the classical value of unity for a hydrogen or fluorine atom. In the Supporting Information, V(r) theory and calculations for H3, HFH, FHF, and F3 are reported for the II T III resonance. Conversion of Excited States of Reactants (R*) into the Excited States of the Products (P*) and the State Correlation Diagram For this section, we need to consider ΦRC ) Φ′RC + µΦ′′RC and ΦPC ) Φ′PC + νΦ′′PC, in which the Φ′′RC of eq 3 is equivalent to eq 26.
Φ′′RC ) (1 - k'k'')(-Φ1 + 2Φ4 + 3lΦ5)
(26)
The corresponding expression for Φ′′PC is given by eq 27.
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TABLE 1: Transition-State Wave Functions ΨTS(II T III) ) ΦRC - ΦPC rXR ) rRY (au) ζ(H)X ) ζ(H)Y ζ(H)R C1 ) -C4 C2 ) +C5 C3 ) -C6 C7 ) -C8 E (au)
HHH
HFH
FHF
FFF
1.88 1.0 1.3 0.26672 0.05177 0.14910 0.09382 -1.60658
2.45 1.04
2.27
3.11
1.21 0.33321 0.16672 0.26879 0.07990 -198.40127
0.43232 0.17518 0.17470 0.04877 -296.75921
0.34141 0.19210 0.18996 0.10420 -99.97474
TABLE 2: MO Parameters and Valencies For VB Structure II at the Transition Statea k′′ k′ l V(ry) V(rx) V(r) V(r)(l ) 0) a
HHH
HFH
FHF
FFF
-0.21757 0.27950 0.62925 0.692 0.297 0.989 0.801
0.00707 0.09498 0.54853 0.869 0.206 1.075 0.9997
0.10154 0.40344 0.29725 0.923 0.103 1.026 0.949
0.06305 0.20205 0.27917 0.961 0.074 1.035 0.983
V(r)(l ) 0) ) V(ry)(l ) 0).
SCHEME 3: ΦRC(µ ) 0) f ΦR{(XR)*(S ) MS ) 1) + Y(S ) -MS ) 1/2} and ΦP{X(S ) -MS ) 1/2) + (RY)*(S ) MS ) 1)} f ΦPC(ν ) 0)
SCHEME 4: (Reactants)* f (Products)*a
Φ′′PC ) (1 - κ′κ′′)(-Φ4 + 2Φ1 - 3λΦ2)
(27)
At the conclusion of the reaction (distance rRY) ∞), the R-Y bond of the reactant-like complex II breaks, to give k′ ) k′′ ) 0 in the MOs φ′ry ) r + k′y and φ′′yr ) y + k′′r and eq 28.
ΦRC(rRY ) ∞) ) Φ′RC + µΦ′′RC ) (1 - µ)Φ1 + 2µΦ4 + l(1 + 3µ)Φ5
(28)
Similarly, at the commencement of the reaction (rXR ) ∞), the X-R bond of the product-like complex III is broken, to give κ′ ) κ′′ ) 0 in the MOs φ′rx ) r + κ′x and φ′′yr ) x + κ′′r and eq 29.
ΦPC(rXR ) ∞) ) Φ′PC + νΦ′′RC ) (1 - ν)Φ4 + 2νΦ1 - λ(1 + 3ν)Φ2
a Because ΦR{X(RY)*(S ) MS ) 1)} and ΦR{X(RY)*(S ) 1, MS ) 0)} are degenerate, as are ΦP{(XR)*Y(S ) MS ) 1)} and ΦP{(XR)*Y(S ) 1, MS ) 0)}, the relevant VB structures of Scheme 3 can be used for (Reactants)* and (Products)*.
(29)
When µ ) ν ) 0.2 and l ) λ ) 0 in the MOs φxr ) x + lr and φyr ) y + λr, linear combinations of degenerate spectroscopic excited states are obtained for the X-R product and the R-Y reactant, according to eqs 30 and 31 for (XR)* and eqs 32 and 33 for (RY)*.
ΦRC(rRY ) ∞) ≡ Φ(Products)* ) 2 CxRrRyβ -
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CxRrβyR - CxβrRyR
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(30)
) 2Φ{(XR)*(S ) MS ) 1) + Y(S ) -MS ) 1/2)} Φ{(XR)*(S ) 1, MS ) 0) + Y(S ) MS ) 1/2)} (31) ΦPC(rXR ) ∞) ≡ Φ(Reactants)* ) 2 CyRrRxβ -
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CyRrβxR - CyβrRxR
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(32)
) 2Φ{X(S ) -MS ) 1/2) + (RY)*(S ) MS ) 1)} Φ{X(S ) MS ) 1/2) + (RY)*(S ) 1, MS ) 0)} (33) In Scheme 3, VB structures are displayed for ΦRC(µ ) 0) f Φ{(XR)*(S ) MS ) 1) + Y(S ) -MS ) 1/2} and Φ{X(S ) -MS ) 1/2) + (RY)*(S ) MS ) 1)} f ΦPC(ν ) 0). However, the |CxRyRrβ| ) Φ{(XR)*(MS ) 0) + Y(S ) MS ) 1/2} and
Figure 3. Schematic state correlation diagram for X• + R:Y f X:R + Y•, with X and Y as equivalent atoms. Calculations are needed to determine whether the Ψ or Ψ* has a maximum or a minimum energy at the crossing point.
|CyRxRrβ| ) Φ{X(S ) MS ) 1/2) + (RY)*(MS ) 0)} components of the ΦRC(µ ) 0) and ΦPC(ν ) 0) do not define (S ) 1 spin) spectroscopic states for (XR)* and (RY)*. Because it is necessary that µ ) ν ) 0.2 in order to obtain degenerate (S ) 1 spin) excited states, the spins of the electrons for the reactant-like and product-like complexes are not specified either in Scheme 4 for the conversion of the excited states of the reactants into the excited states of the products or in Figure 3. Combining Schemes 2 and 4, but without specifying the electron spins for the reactant-like and product-like complexes, we obtain the state correlation diagram of Figure 3 for the radical transfer reaction, when it is assumed that the ΦRC and ΦPC, which are used to construct the ground-state Ψ ) ΦRC + FΦPC,
Further Examples of Valence Bond Structures SCHEME 5: (Reactants) f (Products) and (Reactants)* f (Products)* for X:(-) + R:Y f X:R + Y:(-)
are the same as those used to construct the excited state Ψ* ) F*ΦRC - ΦPC. However, at intermediate stages along the reaction coordinate, excited states Φ*RC and Φ*PC (with different values for the MO parameters k′, k′′, l, κ′, κ′′, and λ and the F*) can replace2 the ΦRC and ΦPC. On numerous occasions, the VB representation of Scheme 5 for the ground-state reaction X:(-) + R:Y f X:R + Y:(-) has been provided.10 The corresponding VB formulation for conversion of the excited states of the reactants of this reaction into the excited states of the products is also displayed in Scheme 5. For this reaction and also for X• + R:Y f X:R + Y•, the electronic reorganization needed for (Reactants)* f (Products)* is essentially the reverse of that for Reactants f Products. Four-Electron Three-Center Bonding Scheme 5 involves a four-electron three-center bonding unit. As is the case for the three-electron three-center unit of Scheme 1, the VB structures for the reactant-like and product-like complexes, each with a fractional electron pair bond and an intermolecular one-electron bond, are increased-valence structures.1a,b,10–12 The bonding electrons are accommodated in the φxr, φ′ry, and φ′′yr MOs of the reactant-like complex and the φ′rx, φ′′xr, and φyr MOs of the product-like complex. The (ground-state) resonance between the increased-valence structures of Scheme 5 generates the variationally best energy for resonance11 between six canonical Lewis structures. Resonance between the two increased-valence structures can help to provide compact VB representations of the ground-state electronic structures for hypercoordinate and nonhypercoordinate systems that involve at least one four-electron three-center bonding unit.1a,b,10–12 (Recent publications for which this approach is relevant for some of the systems considered include those of ref 13 for example.) When the singlet diradical structure is excluded from the canonical Lewis structure resonance scheme, the development of intermolecular oneelectron bonds and “increased-valence” (electronic hypervalence) for the R atom does not occur. The singlet diradical structure is neither a reactant nor a product canonical Lewis structure. It has been omitted from a two “bonding structure” treatment14 of strong hydrogen bonds. Other VB Representations for the Ground-State Reactions In Scheme 6, VB formulations for the ground-state reactions are displayed without the formation of reactant-like and productlike complexes with intermolecular one-electron bonds (i.e., l ) λ ) 0 in the MOs φxr and φyr). Only the reactant and product VB structures and their associated canonical Lewis structures are involved (i.e., excluding structures 7 and 8 of Figure 1 for X• + R:Y f X:R + Y• and the singlet diradical structure for X:(-) + R:Y f X:R + Y:(-)), to give twostep VB formulations. The familiar concerted one-step formula-
J. Phys. Chem. A, Vol. 114, No. 33, 2010 8577 SCHEME 6: Reactants f Products without the Formation of Intermolecular One-Electron Bondsa
a Ψ(covalent) ) Φ1 - Φ4 ) 2|CxRyRrβ| + |CxRrRyβ| - |CyRrRxβ| ≡ 2Φ{(XY)(S ) MS ) 1) + R(S ) -MS ) 1/2} - Φ{(XY)(S ) 1, MS ) 0) + R(S ) MS ) 1/2)}.
SCHEME 7: Familiar VB Representations for Reactants f Products Reactions
tions displayed in Scheme 7 imply that l and λ are not equal to 0 in the MOs φxr and φyr, and therefore, they conceal the formation of the intermolecular one-electron bonds of Schemes 1 and 5 at intermediate stages along the reaction coordinate. Spin-coupled (SC) VB representations for X:(-) + R:Y f X:R + Y:(-) reactions have used one configuration with four three-center nonorthogonal orbitals (SC orbitals) to accommodate the four active-space electrons15 and two singlet-spin configurations. Therefore, the most general type of VB structure that can be obtained from such a treatment corresponds approximately to [X:R:Y](-) for the primary spin coupling scheme. (This structure can also be a type of increased-valence structure; cf. ref 7c.) Depending on the nature of the orbitals, for a symmetric transition state, [X:R:Y](-) can approximate to any of
, and (for the transi-
. tion state of a gas-phase SN1 reaction; cf. ref 15) With extended basis sets, eight SC-type orbitals (four for each structure) can also be used to construct the ΦRC and ΦPC wave ˙ · R:Y](-) and functions for the increased-valence structures [X (-) ˙ [X:R · Y] in Scheme 5. Kekule´ and Dewar Increased-Valence Structures for C6H6 One qualitative VB representation for the delocalized π-electrons of C6H6 involves resonance between the two Kekule´ and three Dewar structures of types 1 and 2 in Figure 4. The Kekule´ and Dewar structures can be stabilized by breaking a C-C π-bond and delocalizing the two electrons into C-C bonding MOs, to give VB structures of types 3 and 4. Increased-valence representations for two three-electron three-center bonding units are thereby incorporated into the VB structures for C6H6. There are six equivalent “Kekule´ increased-valence” structures of type 3 and six equivalent “Dewar increased-valence” structures of type 4. For a three-electron three-center bonding unit, we shall assume that k′ ) k′′ ) 0 in eq 5, to give ΦRC ) Φ1 + lΦ5 and the VB identity
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Harcourt at least partly with the larger spatial separation of the +q atomic formal charges in 4, for 0 < l e ∞. Also, the component Lewis structures 8 and 9 for structure 4 involve more nearest-neighbor bonding than do the component Lewis structures 5 and 6 for 3. As the magnitude of l increases, the importance of the six Dewar increased-valence structures should increase. Although the conclusion that E(Dewar, increased-valence) < E(Kekule´, increased-valence) is in accord with these electrostatic and nearest-neighbor bonding considerations, obviously better calculations (for example, not only better basis sets and choosing the k′ and k′′ parameters of eq 5 variationally) are needed to provide support for the conclusions obtained from this preliminary study. It is also recognized that increased-valence structures for four-center and five-center bonding units can be constructed, but they are not needed for consideration here. Molecular Orbital Treatment of the Interaction of X• with R:Y
Figure 4. Kekule´, Dewar, Kekule´ increased-valence, Dewar increasedvalence, and component co-ionic structures.
TABLE 3: C6H6a W(Kekule´) W(Dewar) energy (au) l in r + ly or y + lr 1T2 3T4 3 4
0.405 0.047 1/6
0.063 0.120 1/6
2.954 2.603 2.711 2.643
0.6 0.9 0.8
Simple three-center MO theory provides a familiar procedure for describing the primary interaction between the radical X• with the substrate R:Y. It involves linearly combining the x orbital with the antibonding MO φry* ) k*r - y. This MO, which is orthogonal to the bonding MO φry ) r + ky when k* ) (k + Sry)/(1 + kSry), is vacant in the separated reactants when the (uncorrelated) wave function for the substrate R:Y is formulated as |φryRφryβ|. As well as interacting (via overlap) with φry*, the x orbital also interacts with φry, and therefore, the threecenter MO ψ ) x + γ1φry* + γ2φry can be constructed. The resulting (S ) MS ) 1/2 spin) Slater determinant is |ψRφryRφryβ|, which, because |φryRφryRφryβ| ) 0, is equivalent to eq 34.
a
To accommodate the two electrons that form the one-electron bonds, each of the six Kekule´ increased-valence structures uses MOs of the type r + ly and y + lr. Three of the six Dewar increased-valence use r + ly type MOs, and their three mirror-image structures use y + lr type MOs.
TABLE 4: Cyclic H6 W(Kekule´) W(Dewar) energy (au) l in r + ly or y + lr 1T2 3T4 3 4
0.432 0.023 1/6
0.045 0.143 1/6
3.049 2.701 2.800 2.714
0.5 0.7 0.6
R
R
|ψ φ
ry
φryβ ) xRφryRφryβ + γ1 φry*RφryRφryβ
| |
|
|
(34)
For the reactant-like complex II, the corresponding wave function, ΦRC ) |φxrRφryRφryβ| with φxr ) x + lr and φry ) r + ky, can be expressed according to eqs 35 and 36.
|φ .
R
xr
φryRφryβ
|
R
R
ry
|
β
ry
R
ry
≡ xRφryRφryβ
R
R
|x φ φ | + l|r φ φ | ) | x φ φ | + kl | r y φ | | + kl|φ * φ φ | /(kk* + 1) )
R
for C6H6. Via this identity, each increased-valence is equivalent to resonance between the Kekule´ or Dewar structure from which it has been derived and three co-ionic Lewis structures (5-7 for 3 and 8-10 for 4). In Tables 3 and 4, we report estimates of elementary STO6G energies and weights for different sets of Kekule´ and Dewartype structures for the π-electrons of C6H6 and the 1s electrons of H6 with D6h symmetry. Each structure has six singly occupied orbitals (a, b, ..., f), and the appropriate S ) 0 spin wave function involves a linear combination of eight Slater determinantal wave functions, for example, those that arise from (a-b, c-d, e-f)type spin pairings. Bond lengths of 1.4 and 1.058 Å, core charges of +1, and orbital exponents of 1.625 and 1.0 were assumed for C6H6 and H6, respectively. The results of the calculations show that whereas resonance between the two Kekule´ and three Dewar structures generates larger weights for the Kekule´ structures, the converse is the case for resonance between the six Kekule´ increased-valence and the six Dewar increased-valence structures. Inspection of increased-valence structures 3 and 4 suggests that the origin of the stabilization of 4 relative to 3 is associated
|
ry
R R
β
ry
R
ry
(35)
β
ry
R
ry
β
ry
β
ry
(36)
Therefore, when γ1 of eq 34 is equivalent to the kl/(kk* + 1) of eq 36, interaction of the X• electron with the antibonding MO of the substrate R:Y is equivalent to the creation of a one-electron X-R bond between the reactants X• and R:Y to form the reactant-like complex II. The theory can be elaborated (via the use of ψ ) x + γ1φ′ry* + γ2φ′′yr*)16 when |φxrRφryRφryβ| is replaced by the Φ′RC of eq 2. Similar types of deductions have been made previously for four-electron three-center bonding units.1b,17,18 Conclusion The VB formulations considered in this paper use increased˙ · R: valence structures for reactant-like complexes (X · R:Y or [X ˙ ](-)), each Y](-)) and product-like complexes (X:R · Y or [X:R · Y of which has a fractional electron-pair bond and a one-electron bond. At the minimal basis set level and using two-center MOs, the ground-state resonance between pairs of these structures
Further Examples of Valence Bond Structures
J. Phys. Chem. A, Vol. 114, No. 33, 2010 8579
gives the variationally best energy for resonance between the component six or eight canonical Lewis structures.1–3,11 Extended basis set orbitals of the types used in the spin-coupled VB calculations of ref 15 can also be used to construct wave functions for the reactant-like and product-like complexes. For ground-state and excited-state reactions that involve resonance between pairs of these structures at intermediate stages along the reaction coordinate, one-electron transfers occur either from an AO into a bonding MO, from a bonding MO into a bonding MO, or from a bonding MO into an AO. Alternative approaches4,13–15 to VB formulations of the reaction of R:Y with either X• or X:(-) do not utilize this compact algorithm to describe how electron transfer can occur as reactants are converted into products.
(b) With φxr ) x + lr, φry ) r + ky in ΦRC ) |φxrRφryRφryβ|, l > 0, k > 0, and ZDO assumed, the resulting expressions for Vrx and Vry are given by eqs A6 and A7.
Acknowledgment. I am indebted to and thank (a) Dr. W. Roso for providing me with his ab initio VB program and (b) Professor Brian J. Duke for program installation and discussion. I have much pleasure in dedicating this paper to Professor Klaus Ruedenberg. His recognition of the role for covalent bonding of the “crucial depression” of the kinetic energy when constructive interference of overlapping AOs occurs has led Wilson and Goddard (Wilson, C. W.; Goddard, W. A. Theor. Chim. Acta 1972, 26, 195. ) to write that Ruedenberg’s work is “one of the more significant developments in the theory of the chemical bond since Pauling’s classic papers.” For nearly 20 years, I taught this theory to third year students of valence at the University of Melbourne, often using section 7 of Ruedenberg’s Rev. Mod. Phys. 1962, 34, 326. as the basis for the teaching.
Appendix
Appendix (1) R-Atom Valence for ΦRC ) |OxrrOryrOryβ| with ZDO Assumption. (a) The ground-state MO configuration for a symmetrical three-electron three-center bonding unit is given by eq A1. It can be transformed19 to give eq A2
ψ(MO) ) (x + kr + y)R(x - y)R(x + kr + y)β ∝ β 1 1 (2x + kr)R(kr + 2y)R x + kr + kr + y 2 2
|
|
{(
R
R
|
) ( R
R
)} |
) ( φxr φry φxr + φxr φry φry ) /2 (A1)
|
β
| |
β
|
≡ -ΦPC + ΦRC
(A2)
in which φxr ) 2x + kr, φry ) kr + 2y, and k > 0. When ZDO is assumed, the R-atom valencies for Ψ(MO) are given by7c eq A3.
Vrx ) Vry ) 6k2 /{(k2 + 4)(k2 + 1)}
(A3)
The maximum value for V(r) ) Vrx + Vry is7c 4/3. The corresponding valence formulas for the ΦRC of eq A2 are those of eqs A4 and A5
Vrx ) 4k2 /{(k2 + 4)(k2 + 2)}
(A4)
Vry ) 4k2(3k2 + 8)/{(k2 + 4)2(k2 + 2)}
(A5)
which give a maximum value of 1.178 for V(r) ) Vrx + Vry (with k2 ) 3.28, Vrx ) 0.3413, and Vry ) 0.8364).
Vrx ) 2k2l2 /{(l2 + 1)(k2 + 1 + k2l2)}
(A6)
Vry ) 4k2(k2 + 1 + 1/2k2l2)/{(k2 + 1)2(k2 + 1 + k2l2)} (A7) When k2 ) 1.16 and l2 ) 0.71, V(r) ) Vrx + Vry has a maximum value of 1.180, with Vrx ) 0.3229 and Vry ) 0.8572. This theory replaces that provided in ref 20.
(2) STO-6G Wave Functions: Method of Calculation. In ref 3, Roso’s ab initio VB program21 was used to calculate the STO-6G wave functions for HFH and FHF. The calculations used fluorine atom best-atom exponents22 and energyoptimized exponents (ζ) for the hydrogen atom(s) and HF internuclear separations (rXR ) rRY at the transition state). The same procedure is used to calculate the corresponding eq 25 type wave functions of Table 2 for H3 and F3. Obviously, an STO-6G basis set is crude by current standards, but the purpose of these STO-6G calculations is to illustrate how electronic hypervalence arises rather than to calculate accurate values of atomic valence. Supporting Information Available: With AO overlap integrals included, the derivation of an equation for V(r) ) V(rx) + V(ry) for ΨTS(II T III), together with calculations, is provided. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) (a) Harcourt, R. D. J. Mol. Struct.: THEOCHEM 1991, 279, 39. (See also: Harcourt, R. D. J. Mol. Struct.: THEOCHEM 1992, 253, 363 for a similar approach to nucleophilic substitution reactions and corrections for the 1991 paper.) (b) Harcourt, R. D. Eur. J. Inorg. Chem. 2000, 1901. (c) Harcourt, R. D. J. Phys. Chem. A 2003, 108, 10324. (2) (a) Harcourt, R. D.; Ng, R. J. Phys. Chem. 1993, 97, 12710. (b) Correction: Harcourt, R. D.; Ng, R. J. Phys. Chem. 1994, 98, 3276. (3) Harcourt, R. D.; Schaefer, K.; Coote, M. L. J. Phys. Chem A 2007, 111, 13278. (4) See for example: (a) Su, P.; Song, L.; Wu, W.; Hiberty, P. C.; Shaik, S. J. Am. Chem. Soc. 2004, 126, 13539. (b) Hiberty, P. C.; Megret, C.; Song, L.; Wu, W.; Shaik, S. J. Am. Chem. Soc. 2006, 128, 2836. (c) Shaik, S.; Hiberty, P. C. A Chemist’s Guide to Valence Bond Theory; Wiley: New York, 2007. (5) Bachler, V. Theor. Chem. Acc. 1997, 96, 223, and refs 55 and 56 therein. (6) Wiberg, K. B. Tetrahedron 1968, 24, 1083. (7) (a) Harcourt, R. D. J. Am. Chem. Soc. 1978, 100, 8060 (Correction: Harcourt, R. D. J. Am. Chem. Soc. 1979, 101, 856). (b) Harcourt, R. D. Aust. J. Chem. 2005, 58, 753. (c) Harcourt, R. D. Aust. J. Chem. 2007, 60, 691. (8) (a) Nalewajski, R. F.; Köster, A. M.; Jug, K. Theor. Chim. Acta 1993, 85, 463. (b) Mayer, I. J. Comput. Chem. 2007, 28, 204. (9) (a) Harcourt, R. D. J. Mol. Struct.: THEOCHEM 2009, 908, 125. The theory in this Erratum replaces that provided in Harcourt, R. D. J. Mol. Struct.: THEOCHEM 2009, 897, 83. (b) See also: Harcourt, R. D. Trends Phys. Chem. 2006, 11, 79. (On p. 81, change -2.9066 a.u. to -2.9063 a.u.). (10) Harcourt, R. D.; Klapo¨tke, T. M. Trends Inorg. Chem. 2006, 9, 11, and refs 3a-c, 4, 39b, 45, and 46b,c therein. (11) Harcourt, R. D.; Harcourt, A. G. J. Chem. Soc., Faraday Trans. 2 1974, 70, 743. when X and Y are equivalent atoms. For non-equivalent atoms, five variational parameters are needed, and these can be chosen as k’ ) κ′, k′′ ) κ′′, l, λ, and F, for example. (12) (a) Harcourt, R. D. The Electronic Structures of Electron-Rich Molecules; The Pauling “3-Electron Bond” and “Increased-Valence”
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Theory. Springer-Verlag: Heidelberg, Germany, 1982; Vol. 30. A 2003 update is available from the author. (b) Klapo¨tke, T. M. In Moderne Anorganische Chemie, 3rd ed.; Riedel, E., Ed.; de Gruyter: Berlin, Germany, 2007 has provided numerous examples of increased-valence structures for four-electron three-center bonding units in particular. (13) Recent publications include: (a) Pierrefixe, S. C. A. H.; Bickelhaupt, F. M. Struct. Chem. 2007, 18, 813. (b) Polo, V.; Gonzales-Navarrete, P.; Silvi, B.; Andres, J. Theor. Chem. Acc. 2008, 120, 341; the expanded valence shell structure X-R-Y with a truly divalent R-atom was also included in the calculations. (c) DeBlase, A.; Licata, M.; Galbraith, J. M. J. Phys. Chem. A 2008, 112, 12806. (d) Sharir-Ivry, A.; Shurki, A. J. Phys. Chem. B 2008, 112, 12491. (e) Sharir-Ivry, A.; Shurki, A. J. Phys. Chem. A 2008, 112, 13157. (f) Braı¨da, B.; Hiberty, P. C. J. Phys. Chem. A 2008, 112, 13045. (g) Woon, D. E.; Dunning, T. H. J. Phys. Chem. A 2009, 113, 7915. (h) Pierrefixe, S. C. A. H.; van Stralen, S. J. M.; van Stralen, J. N. P.; Fonseca Guerra, C.; Bickelhaupt, F. M. Angew. Chem., Int. Ed. 2009, 48, 6469, and references therein. It is noted that four-electron three-center as well as threeelectron three-center bonding units arise in the addition of acyl and silyl radicals to numerous π-systems; see: Schiesser, C. H.; Matsubara, H.; Ritsner, I.; Wille, U. Chem. Commun. 2006, 1067. (14) Humbel, S. J. Phys. Chem. A 2002, 106, 5517.
Harcourt (15) (a) Blavins, J. J.; Cooper, D. L.; Karadakov, P. B. J. Phys. Chem. A 2004, 108, 914. (b) Karadakov, P. B. Chem. Modell. 2008, 5, 312. (c) Cooper, D. L.; Karadakov, P. B. Int. ReV. Phys. Chem. 2009, 28, 2. (16) With a ) 1 + k′k′′, the coefficients γ1 and γ2 can be calculated from {a(k′*) + 2k′′}γ1-{a + 2k′′(k′′*)}γ2 ) al and {a + 2k′(k′*)}γ1{a(k′′*) + 2k′}γ2 ) 2k′l, with k′* ) (k′ + Sry)/(1 + k′Sry) and k′′* ) (k′′ + Sry)/(1 + k′′Sry). (17) Harcourt, R. D. Aust. J. Chem. 1975, 28, 881. (18) (a) Harcourt, R. D. J. Phys. Chem. A 1999, 103, 4293. (b) Harcourt, R. D. J. Phys. Chem. A 2003, 115, 11260. (19) Harcourt, R. D. J. Organomet. Chem. 1994, 478, 131. (20) (a) Harcourt, R. D. J. Mol. Struct.: THEOCHEM 2002, 617, 167. (b) Harcourt, R. D. J. Mol. Struct.: THEOCHEM 2003, 634, 265. For twoelectron three-center and revised four-electron three-center theory of valence, with ZDO assumed, see: (c) Harcourt, R. D. J. Mol. Struct.: THEOCHEM 2005, 716, 99. (d) Harcourt, R. D. J. Mol. Struct.: THEOCHEM 2004, 684, 167. (21) Harcourt, R. D.; Roso, W. Can. J. Chem. 1978, 56, 1093. (22) Clementi, E.; Raimondi, J. J. Chem. Phys. 1963, 3, 2686.
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