6916
J. Phys. Chem. 1991, 95,6916-6918
Some Aspects of the Valence Bond Theory for 6-Electron &Center (or 4-Center 2-Electron) Bonding Richard D.Harcourt School of Chemistry, University of Melbourne, Parkville, Victoria 3052, Australia (Received: March 8, 1991)
When six electrons are distributed among four overlapping atomic orbitals located on four atomic centers A, B, C, and D, four covalent (AB)(CD) and six ionic (AB)-(CD)+and (AB)+(CD)-Lewis valence bond structures may be written down. Two of the covalent structures
-- - -&-- - - -.
0.
Dandg B C D are sometimes omitted from qualitative valence bond descriptions of this type of bonding unit. The consequences of these omissions for the theory of 4-center 2-electron bonding are considered, and illustrated via the results of some nonempirical valence bond calculations for 10 electrons of S2Cl4”. A
B
C
Introduction Attention has been given recently to the phenomenon of e n t e r 2-electron and 6-center 2-electron In refs 1-3, the bonding has been formulated by using both qualitative molecular orbital (MO) and valence-bond (VB) theory. However, as will be shown below, the VB approach presented in these references omits important canonical Lewis structures. These structures were always included in numerous earlier publications, namely those of refs 4 and 5 , in some of which the MO and VB theories were interrelated. In this paper, aspects of the qualitative theory contained in the latter references will be developed further and related to that provided in refs 1-3. The discussion will be restricted to symmetrical 4-center 2-electron bonding, which has been also designated as 6-electron 4-center bonding”aJ*sb*c(cf. ref 6). but it will be illustrated via the rsults of some nonempirical VB calculations for the 10 electrons that form a 6-center bonding unit, Le., a 6-center 2-electron bond, for S&142+,which has been used as a model for Se2142+.3
TABLE I: ChirpninCoulsoll Weights for Lewia VB Shretures, and Energies ( E , au) for Resomoce between (a) 21, (b) 9, and (e) 5 Lewis Structures; (a) Sun of W e i g h for 8 l d c Lewia Structure8 Figure 2a Figure 2b a b C a b C 1 0.106 0.144 0.057 0.109 0.148 0.070 2 (X2) 3 (X2) 4 (X2) 4’ (X2) 5 (X2) 6 (X2)
0.093 0.093 0.028 0.074
0.129 0.129 0.056 0.113
0.094 0.094 0.030 0.073 0.046
0.032 0.439
0.047 0.047 0.128 -1.62
(4
E
(1) Passmore, J.; Klapotke, T. Acc. Chem. Res. 1989, 22, 234.
(2) Burford, N.; Passmore, J.; Sanden, J. C. P. In From Atoms to Molecules, Isorlectronic A M ~ O ~ ULiebman, CS; J. F.,Greenberg, A., Eds.; VCH: Weinheim, 1989; p 53. (3) Nandana, W. A. S.; Passmore, J.; White, P. S.;Wong, C-M. Inorg. Chem. 1990,29,3529. See also: Li, Y.; Wang, X.; Jensen, F.; Houk, K. N.; Olah, 0. A. J . Am. Chem. Soc. 1990, 112,3922. (4) References until 1988 include the following. Harcourt. R. D. (a) Theor. Chim. Acta 1964, 2,437; 1966,4, 202; (b) J . Mol. Struct. 1971, 9, 221; (c) J . Inorg. Nucl. Chem. 1977, 39, 243; (d) Ausr. J. Chem. 1979, 32, 933; (e) J. Am. Chem. Soc. 1980, 100, 5125; 1981,101, 5623; (f) Aust. J . Chem. 1981, 34. 231; (g) J . Mol. Srrucr. (meocHeM) 1985, 122. 235; (h) Ibid. 1988, 169, 193; (I) Qualirative Valence Bond Descriprions of ElecIron-Rich Molecules, Lecture Notes in Chemistry, Vol. 30; Springer-Verlag: New York, 1982, especially Chapters 7 and 8, and Sections 10-2, 11-7, 11-7, 13-2, 13-3, 13-4, and 13-5. Harcourt, R. D.; HILgel, H. M. J . Inorg. Nucl. Chem. 1981,13,239. (k) Skrezenek, F. L.; Harcourt, R. D. J . Am. Chem. Soe. 1984, 106, 3935; (I) Theor. Chim. Acta, 1985,67, 271; 1986, 70, 287; (m) J. Mol. Struct. (meoclreht) 1987, 151, 203. (n) Harcourt, R. D.; Shzenek, F.L.; Maclagan, R. G. A. R. J. Am. Chem. Soc. 1986,108,5403. (5) Harcourt, R. D. (a) J . Mol. Struct. (THeocHeM) 1989,186, 131; (b) Ibid. 1990, 206, 253; (c) In Valence Bond Theory and Chemical Sfrucrurr; Klein, D. J., Trinajstic, N., Eds.; Elsevier; New York, 1990, p 276; (d) J . Chem. Soc., Faraday Trans. 1991,87, 1089. (e) Harcourt, R. D.; Skrezenek, F. L. J . Phys. Chem. 1990, 94,7007. ( 6 ) Carpenter, G. B. J . Chem. Educ. 1963, 40, 385. Extension of the )-center 2-electron theory to that for 6-center 2-electron bonding is easily made,‘..b.i,n
u)
0.058
0.042
0.110
0.423
-1.54
-1.40
0.046 0.125 -1.55
-1.42
-1.61
tures of the general types 1-4 and six ionic ((AB)-(CD)’ and (AB)+(CD)-) structures, two of which are of types 5 and 6. a.
‘A’
a-
D
A
__------- -a e. B
Lewis Valence Bond Structures for &Electron 4-Center Bonding The 6-electron 4-center bonding unit4v5involves six electrons distributed among four overlapping AOs (a, b, c, and d)centered on four atomic centers A, B, C, and D. The qualitative VB theory for this type of bonding (see especially refs 4a,b,e,g-i,5b,c) involves resonance between four (singlet spin) covalent (AB)(CD) struc-
0.129
0.129
e.
A
_---;; -- -, B
C
;’
D
C
a.
D
” ..-.. _-----__
,.e--
C
(3)
(4)
(5)
(6)
‘>D
Examples of structures 1-4 are displayed in Figure 1 for S2C142+,P2F4, and S2042-.7(Structures designated later as 4’ have trans rather than cis locations of the C P , F+,or 0-in 4.) In refs 1-3 and 8, only structures of types 1 and 4 have been included in the VB treatments. However, for the following masons, the excluded covalent structures 2 and 3 may have considerable relevance for the primary descriptions of the bonding. (a) If the formal charge distributions in 2 and 3 are more favorable than are those of 1 and 4, then 2 and 3 will usually be the primary Lewis structures.4g (b) The wave functions q,and q, overlap better with q2and \k3 than they do with each other,**9 and therefore (7) If A and D arc spatially adjacent atoms, as in rectangular Id2+, the formal (-) A-D bond which is present in structure 4 becomes a normal electron-pair bond.’a”” The VB structures for S2042- have been provided previously in refs 4h,i. (8) Kiers, C. Th.; Vos, A. Acta Crysrallogr. 1978, 834, 1499. (9) When zero differential overlap is assumed, and nonneighbor resonance integrals arc neglected. the Hamiltonian matrix element HI4 = 0, whmacl each of H Z 1Ha, , H31,and H,.,is equal to a resonance integral that involves a pair of neighboring a or b and c or d A 0 s . c For the A 0 orientations of Figure 2b, the STO-SG overlap integrals (S = (*,IO )) for SzCI2; &;8:he following values: S I 4 = 0.014. sI2 = * 0.114, sz4 = ~ 3 4
&
0022-365419112095-6916!§02.50/0 Q 1991 American Chemical Society
VB Theory and 6-Electron 4-Center Bonding
The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 6911
F
Because only 10 electrons were included in the calculations, the results are to be used more for the purpose of illustrating aspects of the theory, rather than as necessarily reliable estimates of the (Coulson-Chirgwin") structural weights. Similar calculations4g for six electrons of 142+ suggest that the weights for S2C142+ should be in qualitative accord with at least those that would be obtained from calculations that include all of the valence electrons. In both cases, however, the Slater valence shell A O s are not orthogonal to the core AOse4g
F
Results of Calculations
+
+
+
Figure 1. Lewis valence bond structures of types 1-4 for S2C142+,P2FC
and S2042-.
B\
(a)
JC
(b)
Figure 2. Orientations for a set of 3p AOs for a 6-electron 4-center bonding unit of S$bN. In (a), the axes of the b and c AOs are oriented along the B-C internuclear axis, whereas, in (b), their axes are parallel to those of the a and d AOs. The axes of the a and d AOs are perpendicular to the A-B and C-D internuclear axes, respectively.
the contributions by 2 and 3 help 1 and 4 to These points are illustrated here via the results of some VB calculations (Table I) for a 10-electron 6-center bonding unit of S2C142+,for which four of the six relevant AOs used in the calculations are displayed in Figure 2. Valence Bond Calculatiom The (STO-SG) VB calculations for S Z C I ~were ~ + performed with R o d s programe'O together with the geometry of ref 3. The 10 electrons included in the calculations occupy four 3p A O s located on the chlorine atoms, and two sulfur 3p AOs that overlap to form the SS u bond in the Lewis structure 1 of Figure 1. These are the AOs that accommodate the valence-shell T electrons for two SCI2+monomers. For simplicity, s-p hybridization for the sulfur AOs in the dimer has been neglected. Four of the six AOs are displayed in Figure 2, for two sets of orientations of the sulfur AO's (cf. the 6-electron 4-center bonding unit for N202).5bIn Figure 2b, the monomer 3p?r orientations have been retained for each set of SCI2+moiety A O s in the dimer. Slater 3p exponents for S+and C1, and core charges of +2 for both atoms, were used to evaluate the basic integrals for the six AOs. When the 10 electrons a n distributed among these AOs, 21 (singlet spin) Lewis structures are generated, of which nine are covalent, i.e., they involve (C12S)+(SC12)+type electron distributions. (10) Harcourt, R. D.;Row, W . Can. J. Chcm. IW8,56,1093.
In Table I, the weights for the VB structures are reported. They show that rewnance between structure 1 and those of types 4 and 4' alone provides a rather skewed estimate of the relative importance of these structures in comparison to what is obtained when either the nine covalent structures or all of the structures participate in resonance. When only the nine covalent structures are included in the calculations, the sum of the weights for the four structures of types 2 and 3 is calculated to e x d 0.5. Each of these structures has a larger spatial separation of the formal positive charges (3.33 A) than do either 1 (2.39 A) or the two structures of type 4 (3.06 A). This factor, together with their ability to act as an overlap "bridge" between 1 and structures of types 4 and 4', is responsible for the substantial contributions of structures of types 2 and 3 to the resonance scheme. The latter result has also been obtained for b2+and St+ via either all-valence shell electron calculations, or all-electron c a l c ~ l a t i o n s . ~ g J ~ ~ Therefore, unless either 1 or 4 or 4' is the only important covalent structure, as would be the case for P2F4 (cf. ref 4i with 1 dominant), omission of structures 2 and 3 would lead to an unsatisfactory qualitative VB description of the electronic structure. In all cases, the weight for each of 2 and 3 is larger than the weight for at least one of 1, 4, and 4'. The spatial separation of the positive formal charges in 4', 4.33 A, is larger than the 3.06 A for 4, and this factor is reflected by the much larger weights that have been calculated for structures of type 4'. The contributionsto the resonance scheme of structures of types 2 4 and 4', together with their ionic partners, are a consequence of the delocalization of the A and D lone-pair electrons of structure 1 into the A O s that form the B-C bond in this structure. Such delocalizations can be appreciable if it helps to produce a more favorable distribution of formal charge than what occurs in structure 1. As has been discussed on many occasions previously, the delocalizations lower the B-C bond order and thereby help to lengthen the B-C bond via both the reduction in the extent of B-C bonding4JJ3 and the introduction of B-C antibonding interactions through the inclusion of structures of types 4 and 4' in p a r t i c ~ l a r . ~ h J * ~ ~ * ~ The weights of Table I show little sensitivity to the orientation of the sulfur AOs, but although the orientation of Figure 2a generates lower energies, no reliable conclusion can be drawn as to which would be better if all electrons were to be included in the calculations. Increased-Valence Structures and Spin Pairing On numerous occasion^^*^^^^^^^^^ it has been shown that resonance between the Lewis structures 1-4 is equivalent to use of the increased-valence structure 7. : * I 3
*-e
c * ; (7)
The increased-valence structure may be constructed4either by + spin pairing in a Heitler-London sense (Le., as I...+*k$*Jl I...+*Eda+*dol) the antibonding unpaired electrons of the AB and CD monomers, whose MO configurations are (+J2(+*J1 and (11) Chirgwin, E.;Coulson, C. A. Prof. R. Soc. A 1950, 201, 196. (12) (a) Green, M.;Linnett, J. W . J. Chem. Soc. 1960,4959. (b) Linnett, J. W . J . Am. Chem. Soc. 1961,83,2643. (13) Brown, R. D.;Harcourt, R. D. A m . J . Chem. 1963, 16, 737.
Harcourt
6918 The Journal of Physical Chemistry, Vol. 95, No. 18, 1991
respectively, or by delocalizing an A electron and a D electron of 1into A-B and C-D bonding MOs. Both of these procedures are indicated in 8 and 9, in which the Linnett sym-
($&)'(+*ed)',
x A
o
x B
E
x
;
r2 B
(8)
E
$.,.
(9)
bolismI2 is used to represent the monomer V B structures. Therefore, use of only structures 1 and 4 does not correspond to the VB structure that is obtained through spin pairing of the antibonding electrons. Rather, as will now be shown, the linear combination @ = DITl D4q4.corresponds to a particular form of the linear combination of the (singlet spin) bonding, antibonding and "non-bonding" configurationsq i , qi1, and *ill given by the equation
+
*= cl($.b)2($cd)2($*3'
($*cd)
+ CII($*llb)2($*cd)2($~b)
'
($cd)
'+
~ i i i ~ ( $ o b ) z ( ~ * c d ) 2 ( ~ * ~ b ) ' ( ~ c d+) ' ($*,62)($cd)2($,b)'($*cd)11
(1)
With = u + kb, $*ab = k*a - b, $cd = d + kc, $*cd = k*d - c, and the polarity parameters k and k* related via the local orthogonality requirements that ($&*ab) 1 may be expressed according to eq 2.
Otber Comments On several occasions,q*k it has been calculated that the origin of the rotation barrier for N204and the antiferromagnetismfor Cu(I1) carboxylate dimers is assoCiated much more with resonance between Lewis structures of types 2,3,5, and 6 (with A = D = oxygen, and B = C = nitrogen or copper), than with the presence of a formal A-D bond in 4 or (for the antiferromagnetism)with the presence of a B-C bond in 1. In refs 1-3 and 8, it has been suggested that the contributions of structuresof type 4 to the Lewis VB resonance scheme is responsible for the existence of an eclipsed conformer for S2042-in various crystalline dithionites. However, this effect has been calculatedi4to be insufficient to stabilize the eclipsed S2042- conformer relative to the trans conformer when no counterions are present. Coulombic repulsions between the net formal charges on the oxygen atoms are sufficiently large that they are able to overcome the cis 0 4 3 overlap stabilization that can arise in the eclipsed conformer through rtsonance between structures of types 2, 3, 5, and 6 in particular. We conclude by referring the reader to previous publications~' for examples of increased-valence structures for very numerous systems with 6-electron 4-center bonding units. The increasedwhose orbitals are displayed on the valence structure for S2142+, wer of ref 2, is presented in ref 4i and again in 10. It is obtained
= ( r t c A l l * c d ) = 0,
With (1 - kk*)ClIl = kCi - k*Cii, eq 2 is equivalent to @ = D l q I + D4q4. We thereby obtain eq 3 to relate CI, CIl,and Ciii 10
+
to Di and D4. The parameter k,and hence k* = (k S)/(1 +kS) with S = (ulb) = (c(d),may be determined from a four-structure variational calculation for the dominant configuration Wi = ( I kk*)'{qI - k(Qz q3) k29.,1, When this is done for planar S z C l ~with + the A 0 orientations of Figure 2b, for which (alb) = (cld) = 0.1 19, we obtain k = 1.2 and k* = 1.154. A twostructure variational calculation for 9' and 9, gives D1 = 0.230 and D4 = 0.705. From eq 3, we thereby obtain Cl = 1.17, CIi = 1.04, and Cili = 0.21. Therefore, this calculation shows that spin pairing of the bonding electrons of configuration qi1, as well as spin pairing of the antibonding electrons of configuration q i , may play a very substantial role in the dimerization process when only structures of types 1 and 4 are included in the resonance scheme.
+
+
by spin pairing in a u manner the unpaired r* electron of each 12+ with one of the two unpaired r* electrons of ground-state S The VB structures for the latter two species are respectively' 1.g
+
Acknowledgment. I am indebted to and thank Dr.W. Rmo for his VB program, and Dr.F. L. Skrezenek for installing it. Note Added in Proof. If only the bonding and antibonding configurationsare included in eq 1 and each antibonding MO is not required to be orthogonal to the corresponding bonding MO, then C,,/CI = k/k* = D4/D1. (14) Harcourt, R.D.;Smith, B.; Marsdcn, C. J. Aust. J. Chem. 1984,37, 1553.
