J. Phys. Chem. 1993,97, 12210-12214
12210
Valence Bond Representation for the Hydrogen Atom Exchange Reaction Richard D. Harcourt* and Rickie Ng School of Chemistry, University of Melbourne, Parkville, Victoria 3052, Australia Received: June 22, 1993e
-
+
+
With one hybrid atomic orbital per atomic center, the wave function for the H H2 H2 H reaction a t each stage along the reaction coordinate may be expressed as a linear combination of the wave functions for eight canonical Lewis structures. For non-negligible orbital overlap between the reactants and between the products, the associated ground-state resonance is equivalent to resonance between the increased-valence structures (H0H:H) and (H:H.H), for which the two electrons of the (fractional) H:H bonds of these structures occupy Coulson-Fischer-type molecular orbitals. In this paper, these two types of valence bond representations, and the associated electronic reorganization and curve-crossing diagram, are studied via the results of some a b initio valence bond calculations.
Introduction
+
-
+ +
-
The reaction H H2 H2 H is the prototype example of a radical-exchange reaction X' R-Y X-R Y', and it has been studied extensively by quantum mechanical methods for more than 60 yeami" However, little consideration has been given with regard to how to represent the electronicreorganization that proceeds as the reactants are converted into products. In the present paper, attention will be given to this process, and to the associated state correlation diagram, via wave function formulation and the results of some ab initio valence bond (VB) calculations. If one atomic orbital (AO) per atomic center is used to accommodate the three electrons, then eight canonical Lewis VB structures6J (cf. Figure 1) are needed to describe the reacting system at any stage along the reaction coordinate. Three of these structures are reactant-like (1-3), three are product-like (4-6), and two (7 and 8) participate in the resonance scheme at intermediate stages along the reaction coordinate.6~~If resonance between the eight canonical Lewis structures is used to provide a qualitative VB description of the reaction profile, it is difficult to show succinctly how electronicreorganization proceeds in order to convert the reactants into the products. Associated with this is the question of how to group these eight structures so that the interaction of two diabaticcurves for the Shaik-Pross-type curvecrossing diagrams3-8 is equivalent to resonance between the eight Lewis structures. In a recent paper,' it was indicated that the H + Hz H2 + H reaction could be formulated according to Scheme A.
+
-
(+)
H
(-1
k'H'
Abstract published in Adounce ACS Abstracts, October 15, 1993.
0022-3654/93/2097- 12210%04.00/0
(4
' H ' k H
0
(8)
Figure 1. Canonical Lewis VB structures. @ I = lob4 - lo6c$ @z = lub61, @3 = lac& 0 4 = lcbal- 1~601,@s = Icbbl, @6 = Icaa(, @7 = Ibc& and @a = Iba& in which the presence or absence of a bar over an A 0 in the Slater determinant indicates a fl or a spin wave function. To evaluate the S,, = (@A@,) overlap integrals for eq 23, identities of the type (lobdllcbal) = (abdcba - bca) are used for each pair of Slater determinants.
MO to form an intermolecular one-electron bond, as in structure 11. This structure is able to reorganize its electron distribution
via the transfer of an electron from an H g H c bonding MO into an H A - H ~bonding MO, to afford the VB structure 111. The latter structure forms the products of IV by transferring its HBHc electron into the Hc AO. Structures I1 and 111, examples of "increased-valence" structures,7,9J l-I3 represent intermolecular reactant-like and product-likecomplexes,respectively. Note that the electronic reorganizationwhich obtains for Scheme A parallels that for a more familiar VB representation, namely, that for Scheme B,
-
In this scheme, structures I and IV represent the reactants, in which Coulson-Fischer'O-type molecular orbitals (MOs) are used to accommodate the electrons of the electron-pair bonds. As the reactants approach (linearly) and the H and H2 orbitals overlap, theoddelectronoftheHAatomdelocalizesintoanH~-H~bOnding
(4
but more detail is provided with Scheme A. With the H + H2 H2 H reaction as the example of a radical substitution reaction, the purpose of the present paper is to demonstrate that Scheme A may be used to construct the reaction profile, such that at any stage along the reaction coordinate resonance between I1 and III is equivalent to the vibrational-best resonance between the eight canonical Lewis structures of Figure 1. Consequently, we obtain a compact VB representationthat indicatesclearly how electronic reorganization proceeds as the reactants are converted into the products, and this representation requires only two diabatic curves to construct the curve-crossing diagram. Although accurate calculations of the energy profile that utilizes many AOs per atomic center are now available.14Js the main
+
0 1993 American Chemical Society
Valence Bond Representation for H Atom Exchange
The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 12211
TABLE I: Energy-Optimized Internuclear Separations (RM, RE, au) and A 0 Hybridization Parameters (cf. eq 1). RAb
RE a. ab af
5.57 1.43
99.0 1.43 0.00 0.02 0.02
0.00 0.02 0.02
4.57 1.43 0.00 0.02 0.02
3.58 1.42 0.00 0.02 0.02
2.56 1.39 0.07 -0.21 0.14
1.834 1.834 0.38 -0.25 (0.38)-’
1.43 99.0 50.0 0.02
TABLE II: Coefficients (C’) and Energies ( E au) for the Ground-State Resonance between Lewis Structures 1-8 of Figure 1 (cf. eq 2)‘
CI 0.472 Cz 0.123 C3 0.123
c5 c6 Cl cs c 4
E
-1.6475
0.472 0.123 0.123 -0.0005 -0.0009 -0.0002 0.005 -0.002 -1.6473
6.0 0.471 0.123 0.123 -0.003 -0.003 -0.002 0.011 -0.006 -1.6465
5.0 0.468 0.123 0.121 -0.013 -0.008 -0.008 -0.025 -0.013 -1.6429
3.95 0.438 0.105 0.137 -0.060 -0.022 -0.040 0.059 -0.041 -1.6272
3.668
dbC = b + k‘c,
‘p‘lbC
= c + k“b
(3)
b + K‘U,
gob
= (I + d‘b
(4)
dab=
-
100.43’ @’I1
0.270 0.065 0.125 -0.270 -0.472 -0.065 -0.123 -0.125 -0.123 0.089 -0.089 -1.6098 -1.6475 5
au; REC= 99.0 au. When RAE= 5.96 au and REC= cs = -0,0018, and therefore A cs/c~= m .
The a b initio VB calculations were performed using Roso’s programl619with two (STOdG) 1sAOs per atomiccenter. Three of the AOs (ls’,,, lS’br and ls’,) have H-atom exponents of unity, and the remainder (ls”,,, 1s”b, and ls”,) have the WeinbaumZO Hz exponents of 1.193. At each stage along the reaction coordinate, the three hybrid AOs (a, 6, and c) of eq 1
(1
+ k’k’’)@l+ 2k”@2 + 2k’@3 l(1 + k’k‘93, + 21k’@,
( 1 - k’k’’)(Z@, - @ l - 3 1 ~ ~ )
features of the methodology are illustrated here via the use of one hybrid A 0 per atomic center, each with two 1s components. The theory with two VB structures of types I1 and 111is of course also appropriate for multicomponent hybrid AOs, such as molecularoptimized S C F AOs, a t each stage along the reaction coordinate. We refer the reader to refs 3-6 in particular to obtain detailed VB insights into the origin of the barrier and related properties when VB formulations that do not involve increased-valence structures are used.
Method
= IvabdbCd‘bA - b a b v ’ b c 4 b d E
a RAE= 1.43 c 6 0.0,
1.43 au,
=a
in which I, k‘, k”,A, K’, and are (variational) polarity parameters. Using these MOs, the S = M s= 112 spin wave functions of eqs 5-8 for structures I1 and 111 may be constructed:
RAC 7.0
+ 16, vb, = C + Ab, ‘p.6
Except for the symmetrical transition state, the optimized position of HE was determined for fixed values of RAC. a
100.43
coefficients for the nonnormalized of Figure 1 are reported for the linear combination of eq 2. The electrons of the reactant-like and product-like complexes I1 and 111 of Scheme B occupy the localized MOs of eqs 3 and 4, respectively
(1
+
K’K’’)@~
+ 2 ~ ” @+ 2~ ~ ’ -@ ~ A( 1 + +2xK’@~ K’K’’)@2
Because the same three spatial orbitals are singly occupied in the canonical Lewis structures 1 and 4 of Figure 1 , only the aao afla spin wave function for each of @’n and @’m is needed to construct the wave functions for these structures. We may then write
a = N,,(lSL + aalS’;)
c = N,(lS’:
+ aclS:)
(1)
The terms N,,= (aa2+ 2a,( l ~ h l l s ” + ~ )l)-’/*, etc., were used to construct the S = M s= 1 / 2 spin bond eigenfunctions (3)for the Lewis structures 1-8. The @, are defined in Figure 1 . The hybridization parameters a,,,a b r and a, and the location of HB for various values of the internuclear separation RACwere chosen variationally so that the energy for the linear combination of eq 2 \k =
ccj@j
for which nine variational parameters ( I , k‘, k“, A, K‘, K”, p , p, and u ) arise in eq 1 1. Only seven linearly independent variational parameters are associated with eq 2, and this number may be achieved in eq 11 via several methods. The approach used here involves setting k’ = k” = k and K’ = K” = K , as occurs for free H2. The ground-state values for the seven parameters may then be determined from the Cj of eq 2 by equating coefficients. With d, = C’/C3, we obtain eqs 12-17
1
(2)
p
I
was a minimum for the ground-state resonance scheme. The use of three hybrid AOs with fixed exponents for their component AOs provides an alternative (but nonequivalent) approach to the optimization of the exponents of three unhybridized AOs. The calculations were performed for the reactants and the transition state well as for several intermediate (linear) geometries leading up to the transition state, for which RAB= RBC= 1.834 au. For these calculations, the values of the optimized hybridization parameters are reported in Table I. In Table 11, the Cj
d,, A = d8/d6
hk2 p V
= kd6/K
+ {h(d4- 2d1)+ d , - 1)k + h = 0
= (2k(d, - d,)
= ( 2 ~ ( d-, 1 )
+ I( 1 + k2))/(31(k2- 1))
+ hd6(1 + K2))/(3hd6(K2-1 ) )
(12)
(13)
(15) (16) (17)
12212 The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 TABLE 111: Parameters for Equations 3-12, Calculated from the Ground-State \k with k f = k” = k and K‘ = K” = K’ RAC
100.43
X o ”
I K k
O O 0.133
P V
P P’
7.0
6.0
15.47 0.037 0.0003 0.133 -0.331 -0,333 -0.489 -0.001
3.260 0.089 0.004 0.133 -0.337 -0.333 -0.501
5.0
1.679 0.207 0.016 0.130 -0.344 -0.345 -0.530 -0,008 -0,043
3.95
3.668
1.023 0.431 0.067 0.138 -0.378 -0.371 -0.631 -0.103
0.712 0.712 0.143 0.143 -0.413 -0.413 -1.000 -1.000
100.43b 0 0 0.133 0
Extrapolated;for RAC= 12, 13, 14, 15, and 16 au, X = 6.8,4.5, 3.2, and 1.9. When RAB= 5.96 and 9.94 au, X = +a and -a,respectively. Similarly, when RBC= 5.96, 9.94, and 99.0 au, 1 = + m , -m, and 0, respectively. RAB= 1.43 au; RBC= 99.0 au. 2.5,
**
TABLE IV: Energies (au) for 9, ~ R C \k’pc, , and of Equations 9-11 and 18, with Ground-State Parameters of Table 111’
Harcourt and Ng The values of the parameters reported in Table I11 provide information with regard to how the electronic structures of the reactants and products change as the reaction proceeds, and we now giveconsideration to theseprocesses, with particular attention given to the properties of k and 1. Little variation occurs in the value of k for the H r H c component of the reactant complex during the first half of the reaction. After the transition state has been reached, its value decreases and approaches zero as the reaction nears its conclusion. In contrast, the value of 1 for the reactant complex increases from 0 to 0.7 12 a t the transition state. Beyond the transition state, the value of 1 continues to increase, and a t RBC= 5.95 au23it becomes infinite when calculated from the ground-state 9 of eq 2. However, as we shall now show, its energy-optimized value is zero when an excited state of the products, as well as the ground state, is formed as a spectroscopic state. At the conclusion of the reaction, k = 0 and C5fC4 = 2K/( 1 + K ~ )y. Therefore, eq 9 for the reactant complex I1 reduces to eq 20
RAC ~
O -1.6475 ORC -1.6475 q’pc -1.0856 O* -1.0856 O* 0 -1.0856
= a) = (1 - p)a1+ 2pa4 - 1( 1
@lI(RBc
~~
100.43
7.0
6.0
5.0
3.95
3.668
100.43*
-1.6473 -1.6077 -1.0866 -1.0866 -1.1063
-1.6465 -1.6049 -1.0916 -1.0915 -1.1463
-1.6429 -1.5950 -1.1101 -1.1073 -1.2019
-1.6272 -1.5469 -1.1614 -1.1483 -1.2456
-1.6098 -1.4002 -1.4002 -1.2815 -1.3290
-1.6475 -1.0856 -1.6475 -1.0856 -1.0856
a Calculated from solution of 8 X 8 secular equations for the Lewis structures. RAB= 1.43 au; RBC= 99.0 au.
from which numerical values for these parameters (Table 111) may be calculated. The correctness of eqs 12-1 7 has been checked by using the resulting 1, A, K , and k values to reproduce thegroundstate energies obtained from eq 2. The values of p , p , and v may also be determined independently from the calculated expression for eq 11. For the symmetrical transition state, k = K and 1 = A, p = -1, and Y = p. Energies for the orthogonal excited state \k* of eq 18 \k* = P*\kRC - \kpc
have also been calculated (Table IV). In eq 18, p* eq 19
(18) is given by
(19) P* = ( P ~ I I 1 , I I I+ s I I , I I I ) / ( ~ l I , l I + PSn” with SI^,^^ = ( \ k ~ c l \ k ~ cetc. ) , However, when the LMO parameters for eqs 3 and 4 are calculated from the ground-state Q of eq 2 and used to construct the \k* of eq 18, the energy for \k* is not equivalent to the best variational energy except at the commencement and the conclusion of the reaction (cf. Table IV). In ref 7, it was indicated that Nibler and Linnett21 have also used LMOs that aresimilar to thoseof eqs 2 and 3 toaccommodate one set of three r electrons for the symmetrical anion C3-. These workers assumed that k’= I l k ” = 111 = K’ = 1 / ~ ”= 1fA. The orbitals of eqs 2 and 3 have also been used regularly to accommodate three of the four electrons of increased-valence structures for four-electron three-center bonding units.22
Results and Discussion At each stage of the reaction, either one or both of the covalent Lewis structures 1 and 4 are calculated to be the dominant Lewis structures (Table 11). Near the transition state, polarization of the H : H bonds of the reactants and productsoccurs preferentially in thesense ( H H+ H-) and (H- H+ H), respectively. Theclassical barrier height is calculated to be 23.6 kc8lfmol (cf. exactl4-15 estimates: 9.59 f 0.06 and 9.61 0.01 kcalfmol); because of the smallness of the A 0 basis set used in the present calculations, it is not surprising that these estimates differ considerably. With a larger double zeta + p basis in a VB study, a barrier height of 15.1 kcalfmol has been calculated.4
+ 3p)a5 (20)
and it must correspond to an excited state of the products, whose ground-state wave function is given by eq 21
= = @4 + Y(@5 + @& (21) If it is assumed that @n(Rw = a) of eq 20 represents a spectroscopic excited state, then this wave function must be orthogonal to the @ ~ ~ ( R=B m) c of eq 21. For this condition to be satisfied, we require that @IV(RBC
- P) + 2YPj/{(l + 3P)ZI
1=
(22)
with
andSij= ( etc. TheSijcanonical structureoverlap integrals may be expressed in terms of the A 0 overlap integral s6 . = (alb) to give eq 24 for the x, y, and z overlap integrals
Variationin thevalueofpgivesaminimumenergyof-1.08541 au for @II(RBC= m ) when p = 0.2, Le., when the one-electron bond parameter 1 = 0 in eq 20. We thereby obtain eq 26 for the reactant complex 11,
which represent^^.^.^^*^^ the S = 1 spin state of the HA-HBbond together with the radical state of the H c atom. In accord with what has been proposed previousl? without reference to the wave function for the reactant complex 11, the above treatment shows how the ground state of the reactants of I correlates with an HA-HB (S= 1) + Hc (S= 112) excited state of the products. Similarly, the ground-state for the products of IV correlates with an HA (S = 112) + HB-Hc (S = 1) excited state for the reactants. The VB structures for these excited states of the products and the reactants are V and VI5
(A
A)
+
A
k
(V)
respectively, and therefore the I
-I1
+
(li li) (VI)
V and the IV
-I11
Valence Bond Representation for H Atom Exchange
The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 12213
*
E/a.u.
-l*O
+
1) Hc (S= 1 /2)excited states for the reactants and products (see Appendix). However, Figure 2 is associated with the electronic reorganizations of Scheme A, which shows clearly how the reactants are converted into the products. Other VB Representations Bowen and LinnettZS have used resonance between the nonpaired spatial orbital (npso) structures VI1 and VI11 to h @ H * H
II
'
-1.5
k
-1.6 I I I -1.7 I -4.0 -3.0 -2.0 -1.0
t
(VIII)
represent the electronic structure of the transition state. This is equivalent to assuming that either k" = K" = 0 or k'= IC/' = in the LMOs of eqs 3 and 4. If resonance between these structures is used to represent each stage along the reaction coordinate, the electronic reorganization that is needed to convert the reactants into products is that which is given in Scheme C,
I
-1.4
H * H * k
(VI0
-(Rac
-
3.668)la.u.
I
1
I
0.0
1.0
+(Rac
I
2.0
-
3.0
4.0
3.668)/a.u.+
Figure 2. Diabatic potential energy curves with the ground-state parameters of Table I11 used to construct the wave functions of eqs 9-1 1 and 18: cf. Table IV.
VI processes for the formation and dissociation of the reactant and product complexes, as well as the reaction mechanism indicated in Scheme A, are associated with correlations of spectroscopic states for the reactants and products. In the Appendix, other approaches to the description of the excited states are provided. If values of zero are assigned to the parameters 1.1and v, eq 1 1 will involve seven independent variation parameters when kl # k2 and K I # ~ 2 .For the transition state, this number is reduced to three, and their values are I = 0.712,k'= 0.220,and k" = -0.224. Because k" is negative, the associated +P& and +Po6 MOs of eqs 3 and 4 are antibonding and correspond to the wave functions of eqs 9 and 10. When positive values for these three parameters are determined variationally for either structure I1 or I11 separately using eqs 5 or 7,then the subsequent resonance between I1 and 111does not correspond to the best ground-state energy. For example, a t the transition state with 1.1 = v = 0, this procedure gives k' = K' = 0.24,k" = K" = 0.11,I = X = 0.38, ERC = EPC = -1.5678 au, and E(*) = -1.6043 au (cf. -1.6098 au for the best estimate of E(*)). The reduction from nine to seven independent variation parameters in eq 1 1 may also be achieved by setting 1.1 = I and v = X. When this is done for the transition state, we obtain k' = K' = 2.328and k" = K" = 1.529,with I = X = 0.712. This approach should be the preferred way to proceed for the analysis of the generalized radical-transfer reaction X' + R : Y X : R + Y' with k' # k"and 'K # K" in the R :Y and X : R reactants and products, but it would need to be determined in each case whether or not it yields nonnegative values for all of the k', k", K', K", I, and X bond parameters at each stage along the reaction coordinate. In Figure 2,we plot the diabatic reactant-like and product-like potential energy curves when the ground-state parameters (k,I , K, A, 1.1, and v) are used to construct the \ k ~ and c qp ' c of eqs 9 and 10. At each stage, the energy of the excited state has been calculated from the wave function of eq 18 (with groundstate parameters). The resulting state correlation diagram is equivalent to that described by Shaik et a1.5-6-23.24 when these workers use HA(S = 1/2)+ HB-Hc (S = 1) and HA-HB(S =
-
**
but, unless it is assumed that an electron is transferred directly from the a A 0 into the c AO, the conversion of W to VI11 does not proceed with the facility that obtains for that of I1 to 111. In a recent VB study of the F + HF FH + F reaction, Balint-Kurti et a1.26 used Lewis structures IX-XII,
-
which are equivalent to structures 1,3,4, and 6, to construct their diabatic potential energy curves using their VBSCF procedure. This approach is equiva!ent to assuming that k" = K" = 0 in e q s 3 and 4,giving H-Fand F-H as the VB structures for the reactant and product HF. The resulting electronic reorganization that is associated with the conversion of the reactants into products is indicated in Scheme D,
+
JH7. i
H
+
E
which involves the transfer of an electron from the dk bonding MO of eq 2 to the dab bonding MO of eq 3.
Conclusions Although the VB calculations are elementary in character, they show that when one hybrid A 0 is associated with each atomic center the electronic reorganization that is associated with the familiar one-electron arrow-pushing of Scheme B correlates with the variational-best wave function a t each stage along the reaction coordinate, when this wavefunction is expressed in terms of resonance between two increased-valence structures, as indicated in Scheme A. Scheme A also generates the Shaik-Pross-type curve-crossing diagram which has been provided in refs 4-6. The approach that has been adopted herevia eq 1 1 is the three-electron
Harcourt and Ng
12214 The Journal of Physical Chemistry, Vol. 97, No. 47, 1993
three-center bonding equivalent of eq 9 of ref 27 for four-electron three-centerbonding. The latter equation has now been developed and used on several occasions7~9J~~1* to discuss electronic reorganization for S Nreactions ~ via one-electron delocalizationsand to construct the associated curve-crossing diagram.
Acknowledgment. We thank Dr.W. Roso for his ab initio VB program, Dr.F. L. Skrezenek for installing the program, and The Australian Research Council for financial support. One of us (R.N.) thanks the University of Melbourne for a 1992-93 summer vacation studentship. Appendix
(XVO
(W
(IV)
The VB structures XV and XVI for the excited states were used in ref 7, but their variational wave functions, with 1 * A = 0.88, are not orthogonal to those for the ground states. In order that eqs A2 and A3 may represent spectroscopic states, the orthogonality condition of eq 22 must obtain. With p = 0,this condition gives 1 2 A = x / z < 0 (cf. eq 24). For negative values of 1 and A, eqs A2 and A3 do not correspond to the wave functions for VB structures XV and XVI with one-electron bonds, for which the values of 1 and X are greater than zero.
References and Notes
In refs 4-6, two different approaches were used to provide definitionsof the excited states of the reactants and the products. As was indicated above, one of them corresponds to the use of HA (S = 1/2) H r H c (S = 1) and HA-HB (S = 1) + HC (S = 1/2) to represent these states. The other approach is obtained here by assuming that p = 1 = 0 when k' = k" = 0 and that v = A = 0 when K' = K" = 0 in eqs 9 and 10. Equation 9 then reduces to eq A l ,
+
(1) Truhlar, D. Adu. Chem. Phys. 1977, XXXVI, 141, and references therein. See refs 2-4 below for subsequent VB studies. (2) Howeler, U.; Klessinger, M. Theor. Chim. Acta 1983,63,401; 1985, 67, 485. (3) Maitre, P.; Lefour, J.-M.; Ohanessian, G.; Hiberty, P. C. J . Phys. Chem. 1990, 94, 4082. (4) Maitre, P.; Hiberty, P. C.; Ohanessian, G.; Shaik, S . S.J. Phys. Chem. 1990, 94,4089. ( 5 ) Shaik, S.S.In New Theoretical Conceptsfor Understanding Organic Reactions; NATO AS1 Series C267; (Bertran, J., Csizmadia, I. G., Eds.; Kluwer: Dordrecht, 1989; p 1. (6) Shaik, S. S.;Hiberty, P. C. In Theoretical Models of Chemical BondinK MaksiC. 2.B.. Ed.: SDrinner-Verlan: Berlin. 1991: Part 4.I D. 269. ('IjHarcourt, R. D. THEbCfiEM 199i, 229, 39. (8) Shaik, S.S. J . Am. Chem. SOC.1981, 103, 3692. For additional references, see ref 9 below and: (a) Evans, M. G.; Polanyi, M. Trans.Faraday SOC.1938,34,11. (b) Shaik, S. S.;Pross, A. Acc. Chem. Res. 1983,16,667. (9) Harcourt, R. D. New J. Chem. 1992, 16,667. (10) Coulson, C. A.; Fischer, I. Philos. Mag. 1949. 40, 386. ( 1 1) Harcourt, R.D. (a) THEOCHEM 1988,165,329;corrig. 1989,184, 403. (b) Ibid. 1992, 253, 363. (12) Harcourt, R.D. In Valence Bond Theory and Chemical Structure; Klein, D. J., TrinajstiC. N., Eds.; Elsevier: Amsterdam, 1990; p 251. (1 3) Harcourt, R.D. Qualitative Valence BondDescriptions of ElectronRich Molecules; Lecture Notes in Chemistry; Srpinger-Verlag: Heidelberg, 1982; Vol. 30, p 159. (14) Liu, B. J. Chem. Phys. 1984.80, 581. (15) Diedrich, D. L.; Anderson, J. B. Science 1992, 258, 786. (16) Harcourt, R. D.; Roso, W. Can. J . Chem. 1978, 56, 1093. (17) Skrezenek, F. L.; Harcourt, R. D. J . Am. Chem. Soc. 1984, 106. 3935. (18) Harcourt, R. D. Theor. Chim. Acta 1991, 78, 267. (19) Cui, D.; Harcourt, R. D. THEOCHEM 1991, 236, 359. (20) Weinbaum, S.J. Chem. Phys. 1993, 1, 593. (21) Nibler, J. W.; Linnett, J. W. Trans. Faraday Soc. 1968.64, 1153. (22) See, for example, refs 5, 6, 12a,c, and 15 of ref 9 above. (23) When RAB= 5.96 au. the magnitude of the A 0 overlap integral Sd is less than 0.02, and A = +-. When R A=~9.94au, A = - 0 , and A is negative for 5.97 < RAB< 9.94 au. The associated LMO of q 3,.(pa = c Ab, is then antibondingrather thanbonding. The VBstructureIlIwith the threeelectrons represented as (fra~tional)~*~Jl-~3 bonding electrons will not then represent the product-mplex wave function. For a similar reason, VB structure I1 with three bonding electrons will not represent the reactant+mplex wave function when 5.97 < RE < 9.94 au. Of course, for large internuclear separations theassociated A 0 overlapis negligiblqand therefore the propensity to form the associated one-electron bonds hardly exists. (24) Shaik, S.S.;Hiberty, P. C.; Lefour, J. M.; Ohanessian, G. J . Am. Chem. SOC.1987, 105, 4389. (25) Bowen, H. C.; Linnett, J. W. Trans. Faraday Soc. 1964,60,1185. (26) Benneyworth, P. R.;Baht-Kurti, G. G.; Davis, M. J.; Williams, I. H. J. Phys. Chem. 1992,96,4347. (27) Harcourt, R. D. Aust. J . Chem. 1975, 28,881. '
which has also been derived previously" by consideration of the dissociation products for cP1 at the conclusion of the reaction. In eq A l , @T is given by eq 26 and *S cP4 = lab4 - 1~61, in which the a and b electrons of HA-HB are spin-paired. The resulting energy is -1.3493 au. However, when k'= k"= p = 0 and 1 # 0 in eq 9, we obtain eq A2
(cf. eq 20 with p = 0) to give a lower variational energy of -1.4277 au when 1 = 0.88. The associated VB structure is XV, and the mechanism for the formation and dissociation of the reactant complex I1 is then given by I I1 XV.
--
+
At the commencement of the reaction, the VB structure XVI is the mirror image of VB structure XV,with an energy of -1.4277 au when A = 0.88 in eq A2,
and the formation and dissociation of the product complex 111 from the reactant excited state XVI may be represented as XVI + 111 IV.
-
