6574
J . Phys. Chem. 1990, 94, 6574-6582
The Quantum Mechanical Resonance Energy of Transition States. An Indicator of Transition State Geometry and Electronic Structure Sason S. Shaik,* Eyal Duzy, and Avital Bartuv Department of Chemistry, Ben- Gurion University of the Negev, Beer Sheva, 841 05, Israel (Receioed: October 30, 1989; In Final Form: February 13, 1990)
This paper discusses the quantum mechanical resonance energy ( B ) of three-center transition states having three and four delocalized electrons, such as in atom transfer X-X-X and in polar group transfer reactions (X-A-X)-. It is shown that B is proportional to the energy difference between the HOMO and LUMO of the transition state, and any geometric variation that lowers the energy gap between these two orbitals of the transition state will also decrease its B property. The main causes for possessing a small B are shown to be the geometric looseness of the three-center transition state and its bending away from linearity. The quantum mechanical resonance energy B depends also on the overlap SI2,between the resonating bonding schemes, and decreases as SI2increases. For four-electron/three-centertransition states, of polar group transfer reactions such as SN2,the overlap SI,increases in proportion to the ionicity of the resonating bonding schemes, and therefore B decreases with an increase of the triple ionic character (X- A+ X-) of the transition state. It is shown thereby that B can serve as a probe of transition-state geometry, charge character, and electronic structure. A simple estimation method is devised of quantum mechanical resonance energies of transition states and their ground-state analogues, and the results are compared with recent valence bond ab initio computations of B.
Introduction
There is a search in physical organic chemistry for ways to determine transition-state properties and use them to probe its structure.ls2 Recently there has been some surge of interest in the resonance energy of the transition state. Thus, for example, Malrieu and collaborators3 and Hiberty and collaborators4 have shown that transition-state resonance energies may be computed by valence bond (VB) a b initio techniques for model radical and SN2reactions and that this property depends on the geometry of the transition state.3 From quite a different approach, Eberson: and Lund and collaborators7have utilized experimental strategies to classify and quantify the resonance interaction of the transition state in connection with the outer- and inner-sphere models of the electron transfer theory.*,9 Thus, for example, Lund and L ~ n have d ~ determined ~ the differences in rate constants for nucleophilic substitution and outer-sphere electron-transfer reactions, both measured at identical thermodynamic driving forces, and have quantified thereby the resonance energies of sN2 transition states. The so determined transition-state resonance energies have then been used7* to assess the possibility of a structural continuum in transition-state structure between the outer-sphere and sN2 extremes.iOvll A similar strategy has been ( I ) See, for example, (a) Harris, J. M., McManus, S. P., Eds. Nucleophilicity; Advances in Chemistry Series; American Chemical Society: Washington, DC, 1986. (b) Rappoport, Z., Ed. Reactivity and Selectivity. fsr. J . Chem. 1985, 26, 303-428. (c) Gandour, R. D., Schowen, R. L. Eds. Transition States of Biochemical Processes; Plenum Press: New York, 1978. (2) (a) Jencks, W. P. Chem. Reu. 1985,85, 51 I . (b) More O'Ferrall, R. A. J . Chem. Soc. B 1970, 274. (c) Thornton, E. R. J. A m . Chem. Soc. 1967, 89, 2915. (3) Kabbaj, 0. K.; Volatron, F.; Malrieu. J.-P. Chem. Phys. Lett. 1988, 147. 353. (4) (a) Sini, G.;Shaik, S. S.; Lefour, J . M.; Ohanessian, G.; Hiberty, P. C. J . Phys. Chem. 1989, 93, 5661. (b) Maitre, P.; Hiberty, P. C.; Ohanessian, G.;Shaik, S. S. J. Phys. Chem. 1990, 94, 4089. (5) For a recent review see. Kochi, J . K. Angew. Chem.. Int. E d . Engl. 1988, 27, 1227. (,6) (a) Eberson, L. Electron Transfer Reactions in Organic Chemistry; Springer-Verlag: Berlin, 1987. (b) Eberson, L. Ado. Phys. Org. Chem. 1982, 18. 79. (7) (a) Lund. T.; Lund, H. Acta Chem. Scand. 1988,845,269. (b) Lund. T.; Lund, H. Acta Chem. Scand. 1986, B40, 470. (c) Lund. T.: Lund, H. Tetrahedron Lett. 1989, 27, 95. (8) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155. (9) (a) Hush, N. S. Trans. Faraday SOC.1961,57, 557. (b) Hush, N. S. Prog. Inorg. Chem. 1967. 8, 391. (10) (a) Pross, A. Acc. Chem. Res. 1985, 18, 212. (b) Pross, A,; Shaik, S. S. Ace. Chem. Res. 1983, 16, 363. ( I I ) (a) Bank, S.; Noyd, D. A. J . A m . Chem. SOC.1973, 95, 8303. (b) Walling, C. J. Am. Chem. Soc. 1980,102,6854. (c) Chanon, M. Bull. Chim. SOC.Fr. I I 1982, 213. (d) Lexa, D.; Saveant, J.-M.; Su, K.-B.: Wang, D. L. J . Am. Chem. SOC.1988,110,7617. (e) Lewis, E. S. J. Am. Chem. Soc. 1989, 111, 7576.
0022-3654/90/2094-6574$02.50/0
used by Bordwell and collaborators12to classify the transition-state structure in aliphatic nucleophilic substitution reactions. A general, experimentally based, procedure for determining the resonance energy of the transition state has been recently p r ~ p o s e dand ' ~ applied by Shaik14and Buncel et al.ls to reactions between electrophiles and nucleophiles. The procedure is based on the curve crossing diagram model in Figure la and is illustrated in Figure 1b, for a general chemical process symbolized by R(reactants) P(products).i3s16 It is seen from Figure l a that the forward reaction barrier may be expressed simply as a fraction, J of the vertical diagram gap, GR, minus B as in eq 1. The term B represents the mixing of reactant- and product-like bonding schemes at the crossing point to generate thereby the transition state. Thus, B is simply the resonance energy of the transition state, what we refer to hereafter as the quantum mechanical resonance energy (QMRE) of the transition ~ t a t e . ' ~ . ~ '
-
AEf* = f c R - B
(1)
In principle, the Q M R E of the transition state can be empirically estimated if we find series of related reactions, so-called "reaction families", which obey the linear relationship in Figure 1 b, between the experimental barriers (At!?*and ) the corresponding empirical GR values, which, in turn, can either be m e a ~ u r e d ' J ~ - ~ ~ or estimated from thermochemical cycle^^^^^^ for a variety of reaction types. Having such a linear relationship then the common QMRE value of the transition states is obtained as the intercept ~~~~~~
~~
(12) Bordwell, F. G.; Harrelson, J . A,, Jr. J . A m . Chem. SOC.1987, 109, 8112; J. Am. Chem. SOC.1989, 1 1 1 , 1052. ( 1 3) Shaik, S. S. In New Theoretical Conceptsfor Understanding Organic Reactions; NATO AS1 Series, C267; Bertran, J., Csizmadia, I . G.,Eds.; Kluwer Publ.: Dordrecht, 1989. (14) (a) Shaik, S. S. J . Org. Chem. 1987, 52, 1563. (b) Resonance energies of transition states are available also for the Menschutkin SN2reaction of substituted pyridines, and for electrophile-nucleophile reaction series studied by Kochi et al. (ref 5 above). (IS) Buncel, E.; Um, 1. H.; Shaik, S. S.; Wolfe, S. J . A m . Chem. SOC. 1988, 110, 1275. (16) Shaik, S. S. Prog. Phys. Org. Chem. 1985, 15, 197 (especially p 201). (17) Shaik, S. S.; Hiberty, P. C.; Ohanessian, G.; Lefour, J . M . J . Phys. Chem. 1988, 92, 5086. (18) Solution phase vertical ionization potentials of anionic and neutral nucleophiles are available from photoionization measurements, Delahay, P. Acc. Chem. Res. 1982, 15, 40. ( 19) Solution phase vertical ionization potentials of neutral nucleophiles and substituted benzenes are available in, Watanabe, 1.; Maya, K.; Yoshiki, Y.;Shigero, I. Bull. Chem. Soc. Jpn. 1986,59,907; Nakayama, T.; Watanabe, 1.; Ikeda, S. Bull. Chem. SOC.Jpn. 1988, 61, 673. (20) See for example measurements of vertical charge transfer energy gaps appropriate for a reaction of nucleophilic attack on a carbonium ion, Feigel, M.; Kessler, H.; Walter, A . Chem. Ber. 1978, 1 1 1 , 2947.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 17. 1990 6575
Resonance Energy of Transition States
b
I
\
[Xi--A---X,.]u I
R
I
RC-
I
P
-
Figure 1. (a) A curve crossing diagram for a chemical reaction, R P (after ref 13 and 16). The G's are the electron excitation energy gaps at the reactant and product extremes, respectively. The avoided crossing shown by heavy lines generates the resonant and antiresonant states, 9' and \II*, respectively. fGR is the height of the crossing point expressed as a fraction,J of the gap GR.B is the Q M R E of the transition state and AE? is the barrier in the forward direction. (b) A linear energy plot of the experimental barrier vs the diagram gap, based on eq I .
4e-
Figure 2. (a) An interaction diagram, using the Lewis forms only, showing the generation of the four-electron/three-center transition state and its antiresonant counterpart state for the polar group transfer reaction, Xi:- A-X, Xi-A X : .; (b) The H O M O and L U M O of CH3XC, a typical four-electron/three-center SN2 transition state.
+
-
+
CHART I1
CHART I
the Q M R E and compare the results with recent values obtained by a b initio VB computation^,^^^*^^ with an aim to link the QMRE to fundamental properties of the constituent atoms of the transition state. H:-
*+
H'
+ CH3H'
+ H-H'
-+
+ :H'+ 'H' + :H'-
HCH3
(2a)
H-H
(2b)
H:- + H-H' H-H (2c) of the line.21 Some families have been identified in our ~ o r k ~ ~ J ~ and many more exist in the seminal work of Koch? in the area Theory of charge transfer activation. It is apparent therefore that emThe theoretical problem of the avoided crossing in Figure 1 is pirical Q M R E values will eventually become available, for a classical.26 I t consists of a mixing of two bonding schemes, variety of organic reactions, much the same as these quantities and (P2, to form resonant and antiresonant state^,^^,^^,^' which are are available now for electron transfer reactions from the fitting separated by an energy gap hE(\k*,\k*)as shown in Chart I. The of electron transfer data to the Marcus-Hush and related equaQ M R E ( E ) of the resonant state can be expressed most simply tions.22 in terms of this gap'3t25as eq 3 where SI2is the overlap of the This surge of interest in the Q M R E of transition states, along two bonding schemes in Chart 1. with the potential of eventually using this property to probe transition-state structure, on the one hand, and the growing caB = [ ( I - S12)/2]AE(\k*,'P*) (3) pabilities of a b initio VB computations of Q M R E ' S ~on . ~the ~~~ other hand, face us with a challenge of understanding this tranThe problem now is to associate Si, and hE(\k*,\k*)with more sition-state property, much as we strive to understand it for ground lucid terms which allow both qualitative and semiquantitative state molecules.24 Our purposes in this paper are to lay the considerations to be made. This is done below for two reaction conceptual grounds for understanding transition-state QMRE's classes which involve three-center transition states having four and to explore the relationship between the Q M R E and the and three electrons in delocalization, as in the isoelectronic Kgeometric and electronic structure of the transition state. systems allyl anion and radical. The present paper deals with these purposes by deriving exQ M R E s for Four-Electron1Three-Center Reactions. Consider pressions that allow conceptualization of the Q M R E property, in Figure 2a the (XIAX,)-species which represents a transition for transition states of ionic and radical exchange reactions. These state for the corresponding ionic exchange reaction, where the expressions are then applied for a detailed exploration of the subscripts 1 and r represent respectively the left-hand and geometric and charge distribution dependence of QMRE's of the right-hand X groups. This may correspond to any group transfer model reactions in eq 2 and of the isoelectronic *-analogues of reaction, such as proton transfer or S,2 processes, whose transition the respective transition states in these reactions, the *-components states possess four electrons delocalized over three centers. The of allyl anion and radical. Finally, it is attempted to quantitate corresponding resonant and antiresonant states have been analyzed b e f ~ r e l ~and * ~ shown * to originate in the positive and negative linear combination of the reactant and product pairing schemes, as (21) Of course not every linear correlation should correspond to a "family". demonstrated by the interaction diagram in Figure 2a. and further tests must be utilized to ascertain that the so obtained B is indeed constant and common to all the series. ( 2 2 ) For a few examples see (a) Miller, J. R. New J . Chem. 1987, 11, 83. (b) Miller, J. R.; Beitz, J. V.;Huddleston, R. K. J. Am. Chem. SOC.1984, 106, 5067. (c) Gould, I. R.; Moody, R.: Farid, S . J . Am. Chem. SOC.1988, 110.7242. (d) See a discussion of the resonance interaction in Canon, R. D. Electron Transfer Readions; Butterworths: London, 1980; pp 273-5. (23) (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. (c) Sevin, A.; Chaquin, P.; Hamon, L.; Hiberty, P. C. J. Am. Chem. SOC. 1988, 110, 5681. (d) Sini, G.; Hiberty, P. C.; Shaik, S. S . J . Chem. Soc., Chem. Commun. 1989, 772. ( e ) Maitre, P.; Lefour, J. M.; Ohanessian, G . ; Hiberty, P. C. J . Phys. Chem. 1990. 94, 4082. ( 2 4 ) (a) Heilbronner, E. J . Chem. Educ. 1989, 66, 471. (b) Garratt, J. P. Aromaticity; Wiley: New York, 1986.
( 2 5 ) (a) Levin, G.; Goddard, W. A., 111 J . Am. Chem. Soc. 1975, 97, 1649; (b) Voter, A. F.: Goddard, W. A,, 111 Chem. Phys. 1981, 57, 253. (26) For lucid discussions of avoided crossings see (a) Salem, L.; Leforestier, C.; Segal, D.; Wetmore, R. J . Am. Chem. SOC.1975, 97, 479. (b) Salem, L. Science 1976, 191.822. (c) Devaquet, A. Pure Appl. Chem. 1975, 41,455. (e) Devaquet, A.; Sevin, A.: Bigot, B. J . Am. Chem. Soc. 1978,100, 2009. ( 2 7 ) (d) Salem, L. Electrons in Chemical Reactions; Wiley: New York, 1982; pp 85-86. (28) (a) Shaik, S. S. J . Am. Chem. SOC.1981, 103, 3692. (b) Shaik, S. S . New J . Chem. 1982, 6 . 159. (c) Shaik, S . S . ; Pross, A. J. Am. Chem. Soc. 1982, 104, 2708
6576
The Journal of Physical Chemistry, Vol. 94, No. 17, 1990
For polar group transfers when the A group is more electropositive than the X groups, which is generally the case for most polar group transfers, the resonating bonding schemes at the crossing point of Figure 1 are generated from the corresponding Heitler London wave functions with some triple ion contribution, XI-A+X;, which in turn corresponds to a transition state of the type shown in Chart 11. These resonating wave functions are expressed in eq 4, where the p’s are the orbitals centered on the left-hand (I), right-hand (r), or central (c) fragment as specified by the orbital subscript, spin down is indicated by the usual bar sign over the orbital, and XTI is the mixing coefficient of the triple ion (TI) configuration to both and @2. The S I 2term can be determined using the p r ~ t o c o l in ’ ~Appendix 1. By neglect of the long distance overlap srl,the S I 2quantity becomes eq 4 where the term s is defined in eq 5 and corresponds to the overalp integral of the fragment orbitals of A and X in the transition state in Figure 2a. The mixing coefficient XTI determines the weight of the triple
TI; s = (PrlPc)
-
xT12/(1
(ql~c)
+ hT12)
(6)
In cases where group A is identical or electronically similar to group X, there is another configuration, the so-called “long bond” (LB),29X’A:-’X, which mixes considerably into @, and @2 and affects thereby the SI2quantity, in a similar manner to the triple ion configuration. The S I 2expression for cases that involve a mixing coefficient of the long bond (ALB) configuration can be derived by use of the p r o t ~ c o l in ’ ~Appendix 1 and is shown in eq 7 . Here too the long distance srIoverlap is neglected, while s corresponds to the overlap term between A and either one of the X groups, as specified above in eq 5.
s12=[ 2 s 2 - 2 X s + P ( I
-s2)2+X*s3]/[1
-4Xs+2A2(1 ,2)2
h
1
XLB
