J . Phys. Chem. 1990, 94, 4089-4093 quantitative diabatic curves of VB type, for thermal as well as photochemical reactions. Using basis sets larger than DZ P does not set any conceptual problem; it would simply increase the number of complementary
+
4089
VBFs and therefore the dimension of the CI. Yet the basis sets used in this work provide reasonably good energetics and have therefore been kept in the following paper in which VB curvecrossing diagrams are calculated.
Quantitative Valence Bond Computations of Curve-Crossing Diagrams for Model Atom Exchange Reactions P. Maitre, P. C. Hiberty, G.Ohanessian,* Laboratoire de Chimie Thtorique (UA 506 du CNRS), Universitt de Paris-Sud, 91405 Orsay Cedex, France
and S. S . Shaik* Department of Chemistry, Ben Gurion University of the Negev, Beer Sheva 841 05, Israel (Received: April 20, 1989: In Final Form: September 26, 1989/
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Curve-crossing diagrams are presented and computed for the exchange reactions X' + X-X X-X + X', X = H, Li, by use of a multistructure VB approach. The computations provide the essential diagram quantities C,Jand B. These parameters are 156.8 kcal/mol, 0.37, and 42.4 kcal/mol, respectively, for X = H, and 22.4 kcal/mol, 0.13, and 6.6 kcal/mol for X = Li. The quantitative analyses confirm the qualitative deduction that all these quantities are related to a fundamental property of X, the singlet-triplet splitting AE,,(X-X) of the dimer. It is possible therefore to predict the height of the barrier and the mechanistic modality of the exchange reaction by reliance on AE,,; as PES,decreases in a series the barrier decreases and eventually the X, species is converted to a stable intermediate. The B quantity is the quantum mechanical resonance energy (QMRE) of the X, species. The values of 42.4 kcal/mol for H3 and 6.6 kcal/mol for Li, are computed as energy differences between a variational bonding scheme and the variational adiabatic and delocalized (X-X-X) state.
I. Introduction Curve-crossing VB are general models for discussing reactivity patterns. In the two-curve model 1 (state correlation diagram, SCD)k*bthe reaction profile arises as a consequence of the avoided crossing of two diabatic (or nearly so) curves, one
Knowledge of thef, G, and B quantities is important therefore for predicting and rationalizing trends in the barrier. This approach has proved very useful for the discussion of the barrier problem in SN2,and other electrophile-nucleophile r e a ~ t i o n s . ~ . ~ , ~ In addition, we have already noted that the stability of X,' clusters (n = 3, 4 , 6 ; z = 0, -1; X = monovalent atom or group) correlates with gap size.4 All of these applications have so far relied on qualitative considerations and it becomes essential to test the ideas by rigorous quantitative means which can generate the diagrams and provide insight into the factors that control the variations of the diagram quantitiesf, G, and B. As discussed in the preceding paper,5 the multistructure VB method6 provides the ideal means toward this goal. In this paper, the multistructure VB method is used to generate the curve-crossing SCD for two prototypical exchange reactions, 2 and 3, which represent two mechanistic types of atom exchange
R
!
P
representing the bonding scheme of the reactants, the other that of the products. The barrier of the reaction is given by eq 1 as the difference between the height of the crossing point AEc AE* = AE, - B (la)
AEc = f G (lb) (relative to the reactants' complex R ) and the avoided crossing interaction B. In addition, by appeal to (I), aE, can be expressed as some fraction, J of the diagram gap, G, at the reactant's extreme.2a*b ( I ) Shaik, S. S. J . Am. Chem. SOC.1981, 103, 3692. (2) (a) Shaik, S. S. Prog. Phys. Urg. Chem. 1985, IS, 197. (b) Shaik, S. S. In New Concepts f o r Understanding Organic Reactions; Bertran, J.; Csizmadia, 1. G.,Eds.; NATO AS1 Series; Kluwer: Dordrecht, 1989; Vol. 267, p 165. (c) Pross, A.; Shaik, S. S. Acc. Chem. Res. 1983, 16, 363. (d) Pross, A. Adu. Phys. Org. Chem. 1985, 21, 99.
0022-3654/90/2094-4089$02.50/0
reaction, one passing through a potential barrier and the other (3) (a) Cohen, D.; Bar, R.; Shaik, S. S. J . Am. Chem. SOC.1986,108,231. (b) Mitchell, D. J.; Schlegel, H. B.; Shaik, S. S . ; Wolfe, S. Can. J . Chem. 1985, 63, 1642. (c) Buncel, E.; Shaik, S. S.; Urn, I. H.; Wolfe, S. J . Am. Chem. SOC.1988, 110, 1275. (d) Pross, A. Acc. Chem. Res. 1985, 18, 212. (4) (a) Shaik, S. S.; Bar, R. N o w . J . Chim. 1984,8, 411. (b) Shaik, S. S.; Hiberty, P. C.; Ohanessian, G.;Lefour, J.-M. Noun J. Chim. 1985, 9, 385. (c) Shaik, S. S.; Hiberty, P. C.; Lefour, J.-M.; Ohanessian, G. J . Am. Chem. SOC.1987, 109, 363. (d) Shaik, S. S.; Hiberty, P. C.; Ohanessian, G.; Lefour, J.-M. J . Phys. Chem. 1988, 92, 5086. ( 5 ) Maitre, P.; Hiberty, P. C.; Lefour, J.-M.; Ohanessian, G.,preceding paper in this issue. (6) (a) Hiberty, P. C.; Lefour, J.-M. J . Chim. Phys. 1987,84,607. (b) Sevin, A.; Hiberty, P. C.; Lefour, J.-M. J . Am. Chem. SOC.1987, 109, 1845.
0 1990 American Chemical Society
Maitre et al.
4090 The Journal of Physical Chemistry, Vol. 94, No. 10, 1990
Each excited structure of the diagram retains the electronic character of the ground state which is at the opposite extreme of the diagram. In this manner each curve describes a uniform spin-pairing situation. Thus, in each excited structure there is an infinitely distant radical, X, or X,, singlet-coupled to the X, atom in the X, fragment, while the two electrons in the X, fragment are left uncoupled. For curve I, the wave function of the excited structure reads (3) = N[bl&c(prl - In'Pci%ll where N is a normalization constant. Here the electron spin on X, is half cy and half (being singlet-coupled to X,), while the electron spin on XI is cy. The net result is that the X2 fragment in the excited structure is represented, in terms of the two-electron states of Xz, as a mixture of 25% singlet and 75% triplet. The same holds for curve 11, with interchange of the roles of XI and X,. This definition of the curves allows them to be variationally calculated at any point of the reaction coordinate. An alternative definition of the excited VB structure is, however, possible, in which X, is now a pure triplet. Fuller details of this latter alternative are given in the Appendix. At intermediate points along the reaction coordinate, curve I is an optimized linear combination of the three configurations 4 - 6 which constitute generally the Lewis wave function. In addition to 4-6, there are two zwitterionic configurations 7 and 8 which can mix into curve 1 but only at intermediate points along the reaction coordinate. At the two extremes of the reaction coor**PR
I
-----RC---
I
x,+ X,'
:xi-
:x;
7
yl-
-RC-
-RC-
-
(C)
(6)
Figure 1. (A) A curve crossing diagram for X,' + Xc-X, X,-Xc + X;. G is the diagram gap and fG is the height of the crossing point or the reorganization energy. (B, C ) The diagrams with inclusion of avoided crossing. The states after avoided crossing are shown by dark lines. The avoided crossing resonance interaction is B .
involving a stable intermediate. Since avoided crossings have generally been associated with energy bnrriers, the opposite profiles of 2 and 3 pose a fundamental problem and a challenge for the usefulness of the model. Specifically, (i) can the curves in the model be reproduced rigorously and meaningfully? (ii) Can their avoided crossing reproduce the extreme mechanistic behavior in 2 and 3? (iii) Do the quantitatively determinedf, G, and B parameters follow some fundamental property of the atom? 11. The Qualitative Model The diagram for a three-electron exchange process has been discussed in great detail b e f ~ r e *and ~ ? a~ brief description follows here of the makeup of the curve-crossing diagram. Consider a general three-electron exchange process as in eq 2 , where the subscripts refer to the left-hand side, central, and right-hand side fragments X.
X,'
+ x,-x,
- x,-x,
+ X,'
(2)
Figure 1 A shows the curve-crossing diagram4d without the avoided crossing. Each curve is anchored at a ground state and at an excited V B structure of reactants and products. The ground states of curves 1 and I1 are the usual Lewis structures, where the odd electron is represented by a heavy dot and the bond by a line connecting the bonded atoms. Each Lewis structure is an optimized linear combination of a Heitler-London (H-L) and two zwitterionic structures. These are exemplified in 4-6 for the ground state of curve I, with a line connecting the X,'
x,-x, 4 (H-L)
X,'
:x; x,+ 5
XI'
x,+ :x; 6
heavy dots in 4 symbolizing the two spin-paired electrons of the H-L configuration.
X,' 8
x,+
dinate the latter two configurations belong to charge-transfer states of the general nature X+X2'- and X:-X
