The Driving Force of Addition and Elimination Reactions Clarified through the Hellmann-Feynman Theorem Paul Blaise, Philippe Pujol, and Olivier Henri-Rousseau' lnstitut des Sciences Exactes, Departement de Chimie. Universite d'Oran Es-Senia. Algeria
The driving force of a reaction is a concept which is unclear in the transition state theorv. In order to clarifv this concent. let us consider the saddle point corresponding tu the transition state. The driving force of the reaction will be that which induces the displacement ufthe activated complex before and after the saddle point. Since the transition state involves a maximum in the energetic hypersurface along the chemical pathway, we have
where Ll is the energy of the system in the framework of the Rorn-Oppenheimer approximat~on,and Q # is the reartion coordinate. The whole force which is applied to the nurlei is, therefore, zero so that the concept of driving force seems t o vanish. However, this situation may he the result ofopposiie forces, as it will he shown by using the Hellmann-Feynman theorem (1). The Hellmann-Feynman theorem states that the firrre 3LIlrlQ is equal to the average value of dH/aQ:
Let us now apply these considerations to addition and decomposition reactions. In the addition reactions we have schematically A+R
.c
When A is approaching B along the chemical pathway, the distance hetween the nuclei of A and B is decreasing. Then the nuclear repulsion energy increases along the chemical pathwav. Hence. the nuclear renulsion force tends to drive A awav from H, in the neighhorhoijd ol'the saddle point. ~hereafte;, it mav he concluded that the elertrnn-nuclear attraction knee tends todraw A toward R. This average force may he considered here as the driving force ol'the reaction. It may he emphasized that this conclusion is precisely the intuitive point of view.
t
where &,is the electronic wavefunction of the hamiltonian H. T h e hamiltonian involves four operators H=V,,+V,,+V,,,+7;.
(2)
where Vo, is the electmn-electron repulsion operator.
V,, is the nuclear-nuclear repulsion operator. V"., . .. is the electnm.nuclear attraction imeratur. and T,, is the electronic kinetic energy operator
Of course since v,, and T, do not depend on the nuclear coordinate Q, their derivatives versus Q vanish. Thus, the expression of JH/dQ is
T h e average value of a ~ , , , l ais~rlV,,,/dQ,.where V is the nuclear-nuclear potential energy, because V,, does not operate over the electronic wavefunction. Hence we have
For an enerev extremum this leads to
Let us consider more thoroughly the electron-nuclear attraction force. If this average force favors the approach of A toward B it means that, in the vicinity of the saddle point, the electrons have departed from the nuclei to gather in the central region between A and B. If this statement is true, the forward displacement of the electrons near the transition state must involve an increase of the mean distance between electrons and their original nucleus. One may infer that the average electmn-nuclear potential energy will be smaller, in the region of the transition state, than it would he assumed. As will he shown now, the virial theorem (3)effectively leads to conclusions ( 4 ) in agreement with this statement. According to this theorem there is a relation between the average potential energy ( E p ) and the total energy (1, given by (Ep) = 211 + ZQ,
Of course this result may hold if this extremum c~~rresponds to the energy maximum of the transition state ( 2 ) .There are, therefore, two equal and opposite forces in the transition state. It isvery probable that the antagonism between these two forces is the same a little before and a little after the saddle point. The only difference will be their relative magnitude. Naturally, the force which is the strongest before the saddle point will be the smallest after it. -
' Author to whom correspondence should be addressed.
au a
11)
of course ( E l . ) is expressed as ( ~ P=)i!h~I~~~l$o) + ($dkPl+d
+ Vn,,
18)
where Q,, Pqn,v,.. v,,, and +,,have the same signification as above. Since in the transition state 11 is an extremum, we have A(Er) = + 2 A l l #
(9)
where A l l # is the activation energy of the reaction, and A ( E p ) is the difference in the average potential energy between the transition state and the reactants. Since A l l # is positive, eqn. (9) implies a raising of the average potential Volume 58
Number 8 August 1981
615
energy in the transition state. Obviously, there is not a fortuitous relation between the raising of the average . ~. o t e n t i a l energy in the transition state which fullows from the virial theorem and the average electron-nuclear electrostatic force involved in the activated complex which results from the Hellmann-Fevnman theorem. It may be concluded that the activation energy of the addition reaction follows from the increase in the average potential energy which partially results from the forward movement of the electmn. Since this movement precisely allows the electron distribution to attract the nuclei along the chemical pathway from the reactants to the products, it appears that the activation harrier is connected with the driving force of the addition reaction through the excess of this forward movement. This approach may clarify the physical basis of two ap~roximationmethods which are often successfullv used in the
(6).
In both methods the transition state is approached by the propertres of the wavefunctions of reactants or products. Let us first consider the addition reaction of a nucleophile A with an electrophile B; when A is approaching B, there is in the framework of the frontier molecular approximation, a stahilization energy which results from a mixing between the highest occupied molecular orhital of A and the lowest unoccupied molecular orhital of B. This mixing involves the formation of a charge overlap in the region between A and B. This charge overlap tends, by attracting the nuclei of the entities A and R, to bring them cluser. Clearly, this attracting force is the same as the one which results from the use of the Hellmann-Feynman theorem. Let us now look a t the inverse reaction, i.e.. the d e c o m ~ o sition of C into A and B. The heginning of a dec~rn~ositfion reaction mav he considered as a vibration of the nuclei in which the nuclear displacements belong to the same symmetry as the reaction coordinate. The distortion of the molecule from its equilibrium geometry leads to a raising of the energy. However, this raising is smaller than it would be thouaht hecause there is a s t a h h n g mixing between the electrokc
616
Journal of Chemical Education
ground state and the electronic excited states. This mixing takes dace in such a way that the cost of enerevassociated to he considered to fullow the movement of the nuclei. Now if there are two competitive pathways, the easier will he that which involves the greater mixing and the hetter accompanying of the nuclear displacements hy the electronic distribution. In this approach of the decomposition reaction the nuclear displacements are obviously the inducting process of the reaction; whereas, the electron relaxation is only the leading one which will select one of the competitive pathways. Since in the decomposition reaction the nuclear displacements involve an increase of the nuclear distances, the nuclei may he considered as being influenced by the nuclear repulsion force. If we extrapolate from the onset of the reaction to the transition state, it amears that the driving force of the de.. composition is the nuclear repulsion force, in full agreement with the conclusions of the Hellmann-Feynman theorem. All ahove crrnsiderations lead us to postulate a graduated change of the nuclear coordinate along the chemical pathway when we come from the reactants to the products or vice versa; and this seems to he in agreement with recent a b initio transition state calculati~~ns using sophisticated minimizing processes ( 7 ) . Acknowledgment The authors are grateful to Professor L. Salem for his helpful suggestions. Literature Cited
