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Hardness and bond index profiles of hydrogen-bonded complexes with single-minimum and double-minimum potentials
THEO CHEM Journal of Molecular Structure (Theochem) 309 (1994) 65-77
Hardness and bond index profiles of hydrogen-bonded complexes with singly-minimum and double-minimum potentials S. Nath, A.B. Sannigrahi, P.K. Chattaraj* Department of Chemistry, Indian Institute
ofTechnology, Kharagpur 721 302, India
(Received 2 November 1993; accepted 11November 1993)
Abstract The dissociation reaction, HsN. * ’ HF + H,N + HF and the proton-transfer reaction (F-H. . * Cl)- -+ fF s. . H-CI)have been studied at the HF//6-31G** level in order to understand the progress of these reactions in terms of global and local reactivity parameters. Both the reactions are found to obey the maximum hardness principle. The potential energy curve for the proton transfer reaction passes through a transition state wherein the hardness is at a minimum. In this reaction bond index profiles of the bonds being broken and formed intersect at the point corresponding to the maximum in the potential energy curve. Condensed Fukui functions have been calculated at different stages of dissociation of H3N. . . HF. Variation of reactivity of a particular site towards electrophilic, nucleophilic or radical attack is properly reflected in these values.
1. Introduction
Understanding chemical reactions is a central theme in chemistry. There have been several attempts to provide theoretical justification of the preference for one reaction path over another and formation of only some selected products among the various possible choices. As early as 1963, based on experimental observations, Pearson [l] proposed that a hard base will prefer to react with a hard acid and a soft base with a soft acid to form a stable product. Pearson’s hard-softacid-base (HSAB) principle has been given a theoretical justification within density functional theory (DFT) [2,3]. In DFT, hardness has been defined as [2]
* Corresponding author.
where p is the electronic chemical potential of an N electron system in the presence of an external potential u(r). The expression for or,is given by [4]
(2) Softness is the reciprocal of hardness [5], i.e. 1 s=% In Eq. (2) E is the total energy of the system. However, within the framework of the BornOppenheimer approximation, the above de& nitions of q and p remain unaltered if electronic energy (Eel) is used in place of E. Using finite difference approximation of Eqs. (1)
016~1280/94/$07.00 0 1994 Elsevier Science B.V. Ail rights reserved SSDI 0166-1280(94)03638-2
S. Nath et al.//. Mol. Struct. (Theochem)
66
and (2) expressions of n and p can be written 12-41 as +%+I
-=,+EN-I
=-
2 -/LU
EN-I
-EN+I
2
_
I-A
2
(4)
Z-i-A
2
(5)
where EN, EN- 1 and EN + I are the energies of N, (N - 1) and (N + 1) electron systems, and f and A are the ionisation potential and electron affinity respectively. The HSAB principle is closely related to the principle of maximum hardness (MHP) which states [6] that “there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible”. The principle of maximum hardness under some constraints has been formally proved by Parr and Chattaraj 171.In a subsequent proof by Parr and Gazquez [8] it was concluded that hardness will be extremum where both electronic energy and nuclear repulsion energy reach their respective extremum values. Computations [9-171 on several molecules, molecular clusters and solids show that higher stability is generally associated with larger hardness values. It has been found at the MNDO level of computations [12] that the exothermic exchange reactions of the type AB + CD + AC + BD would produce the hardest possible species as one of the products. It has also been shown for these reactions that the average hardness of the products is greater than that of the reactants. The same has been found to be true for reactions for which an anomeric effect is operative [ 111,A similar observation was made by Nandi et al. [16] for some proton transfer reactions involving silicon hydrides. However, Chattaraj and Schleyer [ 171found that this is, in general, not true for acid-base reactions leading to binary complexes. They have, however, shown [ 171 through state-of-the-art ab initio calculations that the hardness values of the binary complexes of a given acid with different bases increase in the direction of their increasing complexation energies which in most cases match with the direction predicted by the HSAB principle. Hardness profiles for the inversion of NH3 [lo] and PH3 [15] and for intramolecular hydrogen transfer reaction of malonaldehyde [lo] which involves large acti-
309 (1994) 65-77
vation barriers are found to pass through a transition state where hardness is at a minimum and energy is at a maximum. The course of a reaction can also be followed using charge density-related parameters like bond index etc. It seems “that there exists a parallelism between bond index and charge rearrangement in the course of a reaction” [18]. The bond index of the bond being broken or formed in a reaction passes through an inflection point at which the bond index profiles intersect [l&20]. This inflection point closely corresponds to the transition state predicted from the energy profile provided there is a similarity between the bonds being broken and formed. It is known that when a reactant molecule is approached by another, the frontier orbitals play the most significant role since charge distribution in these orbitals determines the stereoseiective behaviour of a reaction [21]. In DFT, the reactivity at different sites of a chemical species can be predicted from local parameters like local softness (S(Y)) [5] which is proportional to the Fukui function (S(u)) [22].
where f
(r)
is defined [22] as
Since in atomic and molecular systems E vs. N plots are discontinuous [23] three different Fukui functions can be obtained by finite difference approximation of Eq. (7) for electrophilic, nucleophilic and radical attack 122-251. Useful information about stereoselective attacks can also be obtained from analysis of the variation of atomic charge in molecules which in turn can be employed to define [24,25] condensed Fukui functions. The condensed Fukui functions have been shown to account for the gas phase basicity of amines, and electrophilic and nucleophilic attacks on several molecules and ions [24-261. In the present investigation we have studied using the ab initio SCF method 1271the course of chemical reactions involving two hydrogen-bonded
61
S. Nath et al.lJ. Mol. Struct. (Theochem) 309 (1994) 65-77
species, namely H,N. . . HF and (FHCI)-. The former one is a normal hydrogen-bonded system with a single minimum potential and the latter is a very strongly hydrogen-bonded system characterised [28,29] by a double-minimum potential corresponding to the conformers (F-H.. . Cl)- and (F ... H-Cl)-. Thus, we have confined our attention to the following reactions. HsN...HF
+ H,N+HF
(F-H..eCl)-
+ (F..&H-Cl)-
(1&, vaiency ( VA) and molecular valency ( VM) are defined [30] by
qA=&aa
(12)
a
(13)
(8)
(14)
(9)
(15)
Reaction (8) leading to the dissociation products takes place along the Cs axis passing through the N. . . H -F moeity. Along this coordinate the reaction does not pass through any transition state which may, however, appear along some other reaction coordinate in the energy hypersurface. In contrast, reaction (9) is associated with a very high energy barrier with (F..-H .+ ‘Cl)- being the transition state (TS). molecular The hardness, chemical potential, valency and bond index profiles have been calculated over the entire course of reactions (8) and (9) to investigate the progress of the reaction. Alternative definitions of n and ~1have been compared. Condensed Fukui functions are calculated at different steps of the NH, . . . HF dissociation to predict sites for the electrophilic, nucleophilic or radical attack. Finally, the reactions have been discussed in the light of the MHP.
where D is the first order density matrix whose explicit form depends upon the scheme of population analysis used. We have used here the commonly used schemes based on Mulliken’s and Lowdin’s population analyses abbreviated as MPA and LPA respectively. The respective density matrices are given by DMvIpA = PS
(16)
DL~A = S”2PS”2
(17)
where P = 2Cf? (C is the coefficient matrix of the doubly occupied MOs) and S is the A0 overlap matrix. The condensed Fukui functionsfkf (for nucleophilic attack), f; (for electrophilic attack) and f{ (for radical attack) are calculated using the following expressions. (18)
2. Method of calculation Ass~ing the validity of Koopmans’ Eqs. (4) and (5) can be written as {2-41 n=
(fLUM0
-
F =
(‘~0~0
+ ~LUMO)/~
EHOM0)/2
.fk
theorem
(10)
(11)
where E denotes orbital energy. In view of their simplicity, Eqs. (10) and (11) have been used for the calculations of n and p. These quantities have also been calcuIated in a few cases using Eqs. (4) and (5). The gross atomic charge (qA), bond index
= qk(N)
-
qk(N
fko=lf21qk(N+1)-qktN-t)j
-
1)
(19) (20)
Here qk is the gross charge on atom k in molecules having N, N + 1, N - 1 electrons. All calculations have been carried out at the HF//6-31G** level. The geometrical parameters for NHs, HF and H3N.. . HF have been taken from literature [31]. Along the dissociation path only the N . . . H distance is varied keeping r(NH) and 8 < HNH fixed at their monomer value and r(H-F) optimised for each value of r(N .+- H). Calculations on (FHCl)- have been performed at r(F ..+Cl) = 3.0, 4.0 and 5.0A. At
68
S. Nath et al./J. Mol. Struct. (Theo&em) 309 (1994) 65-77 -2.60
910
-3.60 6.80
- -023s
I
14Kl
2.00
_3.00
k..
1
600 500 r ( N----H) CA) --c
I
6ffl
Fig. 1. Variation of total energy (E) (----_X hardness fq) I- - -). chemical potential &) (- x - x --I$ molecular valency (FM) c s.f, and bond index I- + - m-f of the H$J.. I HF molecule along the reaction coordinate r(N ~. . H).
each distance the position of H is varied to get the potential energy curve. In order to locate the actual transition state of reaction (91, several r(F +ef Cl) values should have been considered_ The chosen distances are, however sufficiently representative to mimic the nature of the double-minimum potential. In order to determine the condensed Fukui functions and to compare the performance of Eqs. (4) and (5) in the calculation of n and p* the energy and atomic charge of (HsN I ) ~RF)+ and (I-IsN . * . HF)- have been calculated using a UHF formalism [27],
Figure 1 depicts the variation of total energy, hardness, chemical potential, I&, and IN_ _n of H3N. . . HF along the reaction coordinate, r(N - -H), As expected, the energy profile shows a minimum at the equilibrium geometry
of the complex and then rises sharply to reach the dissociation limit, which has been taken here to be r(N - -H) = 6 A. At this distance the binding energy of the complex is only -0.53 kcal mol-’ . The calculated hydrogen-bond energy (- 11.82 kcal mall’) compares favourably with the literature value of - 13.14 kcalmol-’ calculated at the MP4SDTQ level with the 6-3 115 G** basis set and the corresponding ZPE corrected value of -10.35 kcal mol-’ [17’f. The hardness profile along the dissociation path shows that r] is not the maximum at the equilibrium distance corresponding to the lowest value of the energy, as is apparently demanded by MHP. Further, rf passes through a shallow minimum around r(N - -H) $=3 A, although there is no maximum in the energy profile around that point, Both of these features can be understood if the role of the nuclear repulsion potential (V,,) is considered in finding out the conformation with maximum hardness. Although not shown here, the
S. Nath et al./J. Mol. Struct.
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309 (1994) 65-77
69
Table 1 Total energy (_&I, a.u.), hardness (7, eV), relative hardness (nr+ eV), chemical potential (p, eV), relative chemical potential (~~1, eV), molecular valency ( VM), relative molecular valency ( VM)reI and bond index (I,...,) of the N. . H bond along the reaction coordinate rN...n (A) rN...u
-E
7”
-Pa
VM
IN-H
%I b
-A.4
1.6
156.2298
9.021
3.868
3.517
0.083
1.806c
156.2257
8.999
3.797
3.543
0.072
2.206
156.2227
8.873
3.609
3.574
0.057
2.406
156.2203
8.811
3.517
3.580
0.062
3.0
156.2148
8.712
3.269
3.591
0.021
4.0
156.2099
8.734
2.980
3.589
0.001
5.0
156.2083
8.728
2.854
3.595
0.0
6.0
156.2077
8.719
2.800
3.597
0.0
0.301 0.315 0.280 0.293 0.154 0.167 0.092 0.105 -0.007 0.006 0.015 0.028 0.009 0.022 0.0 0.013
1.068 1.203 0.997 1.131 0.809 0.943 0.717 0.852 0.469 0.604 0.180 0.315 0.054 0.188 0.0 0.134
b
( VM )rel b
-0.080 0.782 -0.054 0.808 -0.023 0.839 -0.017 0.844 -0.006 0.855 -0.008 0.854 -0.002 0.860 0.0 0.862
a Hardness (9) and chemical potential (h) have been calculated using Eqs. (6) and (7) respectively. b The first set of numbers denotes values relative to those of HsN t. HF at r(N H) = 6A and the second set of numbers denotes values relative to those of free NHs (n = 8.706eV, p = -5.331 eV, VM = 2.736). ’ Equilibrium value.
electronic energy (E,,) increases monotonically along the dissociation path and the nuclear repulsion decreases in a somewhat similar manner. Neither of them shows any extremum along the reaction coordinate which is a necessary condition for n to attain an extremum value [8]. The opposite trend in the variations of Z&i and V,, causes the hardness profile to pass through a shallow minimum. It is worthwhile mentioning in this context that the importance of V,,, in locating the maximum hardness conformation, has been discussed earlier in the cases of symmetric and asymmetric vibrations of molecules [9,15]. It has been found that in the latter mode V,, remains virtually constant and hardness is a maximum at the equilibrium geometry where Z&t is a minimum. However, in symmetric vibrations, V,, does not remain constant and neither Z&tnor r] becomes an extremum at the equilibrium state. The chemical potential (p) increases and tends to attain a constant value at the dissociation limit (Table 1). The fact that peues < z+,N...HF), indicates that the complex at the equilibrium geometry
has a greater tendency to accept electrons than at the infinitely separated monomer limit. The increase in molecular valency ( VM) is ve:y sharp at the beginning and from r(N . . . H) M 3 A onwards the change becomes rather negligible. Along the dissociation path the change in VM is due to changes in ZN.,,u and ZnF. The former decreases and the latter increases with increasing r(N . . . H). Upto r(N.. . H) x 3 A, the increase in In-F outweighs the decrease in ZN,,,H and Vj increases. Beyond this point ZN...n is negligible and ZH_F remains virtually constant. The variations of VM and n relative to that of NH3 and of (HsN . . . HF)oo (not shown in the figures) follow the same pattern and can be explained using the above arguments. The nature of the variations of n relative to those of NH3 and (HsN... HF)oo implies that n(complex) > r](complex) > n(HsN . . . HF),, rl(NHs), and which are in accordance with the HSAB theory and MHP (Table 1). The bond index profile (Fig. 1) of the N-H bond shows that it is somewhat erratic around
S. Nath et al./J. Mol. Struct.
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309 (1994) 65-77
Table 2 Variation oftotal energy (E, a.u.), hardness (q, eV), chemical potential (p, eV), molecular and H-Cl (ZH_& bonds with increasing F-H distances @(F-H), A)”
valency
( VM),and bond indices of F-H (&,)
r(F-H)
-E
9
0.8 0.8 0.8
559.5479 559.5389 559.5309
9.597 7.768 6.994
5.320 3.854 3.325
0.981 0.938 1.195
0.878 0.906 0.917
0.108 0.034 0.004
0.947 0.947 0.947
559.5706 559.5578 559.5477
9.498 7.435 6.555
4.929 3.387 2.822
0.877 0.841 0.824
0.727 0.797 0.819
0.146 0.044 0.005
1.0 I.0 1.0
559.5668 559.5519 559.5408
9.437 7.275 6.355
4.735 3.854 2.596
0.854 0.819 0.801
0.681 0.768 0.795
0.165 0.049 0.006
1.2 1.2 1.2
559.5368 559.5087 559.4929
9.263 6.606 5.502
3.912 2.222 1.633
0.838 0.791 0.769
0.540 0.703 0.757
0.270 0.079 0.010
1.5 1.5 1.5
559.5041 559.4407 559.4115
9.216 5.952 4.383
3.510 0.865 0.264
0.920 0.843 0.791
0.372 0.631 0.754
0.519 0.176 0.030
1.7 1.7 1.7
559.4907 559.4109 559.3665
9.000 5.904 3.931
4.239 0.220 -0.466
0.978 0.891 0.818
0.296 0.549 0.732
0.683 0.284 0.067
2.0 2.0 2.0
559.3825 559.3958 559.3189
8.894 5.802 3.738
5.167 0.217 - 1.260
1.042 0.931 0.880
0.251 0.386 0.637
0.853 0.478 0.189
2.7 2.7 2.7
551.6759 559.4472 559.3004
7.287 6.407 3.494
4.590 3.245 - 1.027
0.887 0.942 0.926
-0.017 0.097 0.591
0.826 0.847 0.263
’ Three values in each column
correspond
to fixed F.
P
Cl distances
r(N . . H) x 2A and seems to pass through an inflection point around r(N . . H) M 3 A which may be reminiscent of the minimum n value at this point. Some representative values of various quantities calculated for (FHCl)) are given in Table 2. The potential energy (PE) curves of this species resulting from the variation of energy with respect to r(F-H) are shown in Fig. 2 for r(F . . . Cl) = 3, 4 Dand 5 A. The PE curve at r(F.. . Cl) = 3 A (it is less than the sum (3.15 A) of the van der Waals radii of F and Cl) passe: through a single minimum. At r(F . . . Cl) > 3.15 A, i.e. at 4.0 and 5.OA the PE curves exhibit the doubleminimum characteristics. In the last cases the TS, (F..O. H. . . Cl)- corresponds to r(F-H)=2.0 A and 2.5 A respectively. ’
.
VM
IF-H
[H-Cl
of 3 A, 4 A and 5 A respectively.
The hardness profiles (Fig. 3) at r(F . . . Cl) = 4 A and 5A pass through a broad minimum in each case corresponding to the TS. The positions of the minima are however not exactly identical to those of the maxima in the energy profiles since the positions of respective extremum in E,i and I’,, profiles (not shown here) do not coincide. As in the case of HsN . . . HF, the hardness is not a maximum at the equilibrium geometry owing to well-known reasons. However, the n value corresponding to the minimum (9.498eV) in the PE curve at r(F.. . Cl) = 3 A is considerably greater than the respective value at longeroF. . . Cl distances (7.275 eV at r.(F . . . Cl) = 4A, and 6.555eV at r(F.. . Cl) = 5 A). It has been noted earlier [28] that the energy of activation required for reaction (9) is less than that for the reverse
S. Nath et al./J. Mol. Struct.
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309 (1994) 65-77
- 0.00 I
I
I
-0.10
I
I I
I
-02c l-
t
I
c ?
a
=: -0.3c z ul + w -
/ \
-Odd
\
I
/’
-05
-0% O0.00
Fig. 2. Variation
of total energy
I
I
I
I
2Gl
3.00
L.00
500
r
( F-H I(%) --c
I
1.00
(E) of (FHCl)-
with r(F-H)
reaction which agrees with the experimental observation [32] that in the ion-molecule exchange reaction, F- + HCl = HF + Cl-, the proton is transferred to the halide ion with lower atomic number. The hardness profile of this reaction shows that r](F-H . . . Cl)- decreases by a greater margin than n(F . . . H-Cl)while forming the activation complex (F-..H . ..Cl)- with minimum hardness. It follows from the MHP that a system in equilibrium with maximum hardness resists any tendency of lowering in hardness value and consequently the process of forming the activation complex from (F.. . H-Cl)will be easier than that from (F-H. . . Cl)-. In agreement with this, both the hardness profile and the PE curve show the preference of the backward reaction to occur. It can also be noticed from Fig. 2 that the energy barrier is higher at r(F . . . Cl) = 5 A than at 4A. This tallies with the observation that the hardness profile corresponding to the formation of the activation complex is deeper at r(F . . .Cl) = 5A than at r(F...Cl) =4A. It again follows from the MHP that the reaction
at r(F
Cl): 3 A, (-
. - . -); 4& (-),
5 A (-
-
-),
path which has a higher activation barrier and in consequence lesser reactivity will form a TS complex having a lower hardness value. It may be mentioned in this context that the preference of reaction path can also be explained in terms of activation hardness defined by Zhou and Parr [33]. The chemical potential for the system is positive over the entire region for r(F . . . Cl) = 3 and 4 A (Fig. 4) and attains a small negative value at the transition state for r(F . .. Cl) = 5 A. The positive p values imply that (FHCl)-, already a stable closedshell anion, does not have any tendency to accept electrons. However, the TS at r(F.. . Cl) = 5 A seems to behave differently. The Vhl profile (Fig. 5) at r(F es. Cl) = 3 A shows the pre!ence of0 one deep minimum. At r(F . . . Cl) = 4 A and 5 A the presence of another minimum of smaller depth can be noticed. However, the overall variation of VM in the process is rather small. The minimum value of Vhl near the equilibrium geometry emphasises the importance of electrostatic interaction in the formation of (FHCl)-. Moreover, it is interesting to note from
72
S. Nath et al/J.
-\ 7.00
\ \
Mol. Struct. (Theochem)
\
\\ \\
t q(QV)
\
309 (1994) 65-77
/ P /I
J
! : I
\ I
\
ml
\
:
\ \
3oc i
0
I \
/
\ __--
\
\_ ,I’
I
I
2.00
300
r(F-H)(I%) Fig. 3. Variation
of hardness
(7) of (FHCI)-
with r(F-H)
‘\ 4.00
\\
\\
\
!
t
Cl):
3 A, (- .
- . -); 4 A, (-),
!
/
‘\.J
/I
5A (-
- -).
/
I /
i
\ \
I
\ \
1
$eV
-
at r(F.
\ 2.00I-
500
&OO
!./‘-
\
I
I
I
1.00
\ \
\
\
o*ot)\ \
- 2.0( l-
IO o-a
I
l-00
\
/
/
/
1-l’ I
I
2.00
390
I
I
LB.00
500
r (F-H)(%)Fig. 4. Variation
of chemical
potential
(p) of (FHCl)-
with r(F-H)
at
r(F
Cl):
3 A, (-
* - . -); 4A, (-),
5 A (-
- -).
S. Nath et al./J. Mol. Struct.
lo
Fig. 5. Variation
of molecular
valency
(Theochem)
1
I
l%O
200 r(F-H)( ii) -
( VM) of (FHCl)-
I
with r(F-H)
the V, and p profiles at r(F . . . Cl) = 4 A and 5 A that these quantities reach their respective maximum and minimum values around the TS geometry. The variation of bond indices of the bonds being broken (F-H) and formed (H-Cl) along the reaction coordinate has been presented in Fig. 6. The nature of the curves is almost identical for all three F. . . Cl distances. At the F. +. Cl distances of 4 A and 5A, the Zr_, and Zu_cr curves intersect at a point which corresponds (within 0.05-o. 1 A) to the maximum in the respective potential energy curves. This agrees with earlier SCF and INDO/MCSCF calculations [18-201 on systems where the bonds being broken and formed are similar. It has been observed that for thermoneutral reactions the two bond indices are equal at TS in accordance with the principle of conservation of bond indices [34] and Hammond’s principle [35]. However, for exothermic and endothermic reactions these bond indices should not be the same at the saddle point. This has been found to be valid for several types of exchange reactions studied by Lendvay [ 18,191. In the work of Maity and Bhattacharyya [20] inflec-
13
309 (1994) 65-77
I
3.ocl
at r(F
Cl):
&OO
3 A, (- .
1 500
- . -); 4A, (-),
5 A (-
- -).
tion points in the bond index profiles of N-H and O-H bonds in the isomerisation of HNO to NOH are found to coincide with the actual saddle point of the reaction as obtained from the corresponding PE curve. Since this reaction has been found to be an endoergic one, a greater value of ZNHthan ZOH (as is found from Table I of Ref. 20) at the TS seems to be against Hammond’s postulate. We would like to clarify that the intersection point of two bond index profiles around respective inflection points should not be used to locate the saddle point in reactions other than thermoneutral ones because the bond indices will have same value at the TS only for thermoneutral reactions according to Hammond’s principle. It is however, gratifying that in the presently studied reaction (9) of (FHCl)-, which is also an endoergic one, the bond index values at TS show that ZHcl is slightly greater than ZFH (Table 2) which is in accordance with Hammond’s principle [35]. It is interesting to note that the ,,minima in hardness profiles at r(F . . . Cl) = 4A and 5 A (Fig. 3) lie towards the product side which may also be thought of as a consequence of Hammond’s principle.
14
S. Nath et al./J. Mol. Strucr. (Theochem)
r(F-HI
Cli,
309 (1994) 65-77
-
Fig. 6. Variation of bond indices of bonds being broken (I,_,+) .. and formed (ZfI_cI) of (FHCI)- with r(F-H) at r(F (- * - ‘ -); 4‘4, (-), 5A (- - -). -
The condensed Fukui functions calculated for I&N. s’ HF at its equilibrium geometry, dissociation limit and an intermediate point, have been presented in Table 3. Energies of neutral HsF . v- HF and the corresponding positively and negatively charged species along with n and ,LL values calculated from definitions (4) and (5) (using vertical I and A values) have also been reported. The relative values of 7 and ,u obtained using these equations follow the same order, although the numerical values differ owing to the iimitations ofthe Koopmans’ theorem which has the well-known tendency to overestimates and A values. The three sets of condensed Fukui functions revea1 that the value off- is largest for nitrogen which means that due to the presence of a lone pair of electrons it will be the most favourable site for electrophilic attack. The hydrogens attached to nitrogen have a high f+ value and should be potent centres for nucleophilic attack. This agrees with the observed high dipole moment of NHs. The value of ff is very small for hydrogen-bonded hydrogen since some of the lone pair density of
. Cl): 3A,
nitrogen has been transferred to hydrogen due to the inductive effect of fluorine. Nitrogen shows a greater tendency to bind electrophilic reagents (fis higher) than fluorine. In the course of the dissociation reaction, as HF moves away from NH3, condensed Fukui functions show expected trends. The propensity for electrophilic attack on nitrogen increases as r(N . a. H) increases because the lone pair of electrons becomes relatively free. The lower f+ value of the three hydrogens attached to nitrogen and the higher f’ value of the hydrogen engaged in three-centre hydrogen-bonding interaction at r(N sI -H) = 2.206A is in confo~ity with the classical chemical concept. As the hydrogen-bonded hydrogen moves away from the lone pair of nitrogen, it becomes a better nucleophilic centre and hydrogens attached to nitrogen become less prone to nucleophilic attack. Howeve:, near the dissociation limit, at r(N. . . H) = 6.0 A, this trend of the f' of hydrogens gets reversed which may be due to the localisation of negative charge on NH3 in the [NH,. . . HF]- species as predicted by the population analyses (MPA and LPA).
S. Nath et al./J. Mol. Struct.
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15
309 (1994) 65.77
Table 3 Energies (E, au.) of (H3N. . HF)+, HsN. HF and (HsN I HF)- species, hardness (n, eV), and chemical potential (p, eV), and condensed Fukui functions for the HsN. . HF molecule along the reaction coordinate r(N H) Atom r(N.
-Ea
li”
+C
Pb
fk
7.752
2.857
N
H’d H” d F .H)
7.620
0.4911 0.6068 0.1447 0.0952 -0.0577 4.47 x 10-r 0.1326 0.1030
-0.4138 0.1322 0.4410 0.2641 0.0549 0.0419 0.0364 0.0341
0.5116 0.6581 0.1494 0.0972 -0.0495 -0.0118 0.0898 0.0618
-0.3948 0.1601 0.4648 0.2799 -9.04 x 10-s -5.52 x lo-’ 9.05 x 1o-3 5.52 x 1O-3
0.5216 0.6932 0.1595 0.1022 -0.0118 -7.14 x 10-s 0.0118 7.14 x 10-s
0.0293 0.3687 0.2967 0.1814 -6.60 x 10-r 0.0188 0.0845 0.0686
2.641
N
H’ H” F .H)
-0.4324 0.1307 0.4487 0.2676 0.0511 0.0331 0.0364 0.0342 =2.206/i 155.8456 156.2227 156.0397
r(N.
fi”
H) = 1.806d 155.8358 156.2257 156.0458
r(N.
fiC
0.0489 0.3951 0.2952 0.1806 2.70 x lO-3 0.0150 0.063 1 0.0479
=6.0/i 155.8630 156.2083 156.0021
7.496
-1.840
N
H’ H” F
0.0634 0.4266 0.3121 0.1910 -0.0142 -6.33 x IO-’ 0.0142 6.33 x 1O-3
a Three values correspond to the energy of (H3N.. HF)+, H3N.. HF and (H3N . HF)- species respectively. b n and p values are calculated from Eqs. (4) and (5) respectively. ‘The first and second values off:, f ;, f e have been calculated by using MPA and LPA gross charges respectively. d H’ denotes hydrogen attached to nitrogen in NH3 molecule and H” denotes hydrogen-bonded hydrogen in H3N. . HF.
The reactivity trend of the molecule along the dissociation path as obtained from the study of condensed Fukui functions remains the same if either of the MPA and LPA schemes is used. Note that the radical attack on N and H’ centres are, however, predicted differently in these population analysis schemes. The numerical values of
Fukui functions obtained from LPA are lower than those obtained using MPA since LPA overestimates covalent interaction [36]. This overestimation leads to an unusual result in the case of radical attack. While in MPA scheme the radical attack is favoured at three hydrogen atoms, the nitrogen atom is preferred in the LPA scheme. In
76
S. Nath et a1.l.J.Mol. Struct. (Theochem) 309 (1994) 65-77
general, anywhere along the dissociation path f &, fi and_& (f& in LPA scheme) possess the largest values. Since the Fukui function is a normalised local softness (Eq. (6)) a soft nucleophile or a soft radical will prefer to attack at the hydrogen atom of NH3 and a soft electrophile will prefer to bind to the nitrogen atom. Since the hard-hard interaction is supposed to be charge controlled [37] as opposed to the frontier-controlled soft-soft interactions and the total gross charge is the largest (for all f(N . . . H) distances and for both LPA and MPA schemes) on the nitrogen atom, any hard species or hard end of a species would prefer to attack the nitrogen site of the H,N . . . HF. Overall consideration of the two reactions shows that both of them obey the MHP. The formation of HsN.. . HF is exoergic to the extent of 11.82 kcal mol-’ and is accompanied by an increase in n of 0.28eV. In general n decreases from its equilibrium value along the dissociation path which is accompanied by an increase in energy as well as chemical potential. Note that both definitions for n and p (Eqs. (4) (5), (10) and (11)) provide identical trends. Thus, the MHP has been found to be valid although p is not strictly a constant which was a constraint in the Parr-Chattaraj proof [7]. The energy profile of the (FHCl)) ion shows that the minima can be approached from the TS (F...H . ..Cl)- in two ways. Ab initio calculations [28] as well as experimental results [38] show that the formation of the species in which fluorine is closer to hydrogen will be the favourable process since [F-H . . . Cl]- is associated with the minimum energy and the maximum hardness [7]. This is illustrated below u$ng E and n values from Table 2 for r(F...Cl) = 5A. [F...H-Cl]-
[F . . . H . . . Cl]-
going from one species to the other hardness decreases and vice versa which is in conformity with the MHP. Chattaraj and co-workers have noticed the validity of the MHP in various static [ 15,17,39] and dynamic [40] situations.
4. Conclusion This work presents an analysis of chemical reactions associated with single and double minimum potentials in terms of the HSAB theory and MHP. For reactions passing through a TS, the hardness profile is found to pass through a minimum. The greater well depth in the hardness profile is related to a lesser reactivity as has been the case with a greater activation energy barrier. The nuclear repulsion potential is very important in locating the state with maximum or minimum hardness. Hammond’s principle has been discussed from energy, bond index and hardness profiles. The study of condensed Fukui functions for HsN . . . HF dissociation gives general information about gradual accumulation or depletion of charges on atoms in a molecule which in turn determines the centre for the nucleophilic or electrophilic attack in a chemical reaction. Comparison of hardness values of reactants and products show that reactants proceed in a direction in which the product with maximum hardness is formed which is in agreement with the MHP.
5. Acknowledgement
PKC would like to thank the Council of Scientific and Industrial Research (CSIR) for financial help.
E - 559.408 a.u. + -559.293 a.u.
n
4.45eV
3.74eV + [F-H.
. . Cl]-
-559.548 a.u. 6.55 eV Comparison of n values of different species in the above reaction shows that if energy increases in
6. References [l] (a) R.G. Pearson, J. Am. Chem. Sot., 85 (1963) 3533. (b) R.G. Pearson (Ed.), Hard and Soft Acids and Bases, Hutchinson and Ross, Stroudsburg, PA, 1973. [2] (a) R.G. Parr and R.G. Pearson, J. Am. Chem. SOL, 105 (1983) 7512.
S. Nath et al./J. Mol. Struct. (Theochem)
(b) P.K. Chattaraj, H. Lee and R.G. Parr, J. Am. Chem. Sot., 113 (1991) 1855. [3] (a) R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. (b) P.K. Chattaraj and R.G. Parr, in K.D. Sen and D.M.P. Mingos (Eds.), Chemical Hardness, Springer, Berlin, 1992, p. 11. (c) P.K. Chattaraj, J. Indian Chem. Sot., 69 (1992) 173. (d) M.K. Harbola, P.K. Chattaraj and R.G. Parr, Isr. J. Chem., 31 (1991) 395. [4] R.G. Parr, R.A. Donnelly, M. Levy and W.E. Palke, J. Chem. Phys., 69 (1978) 443 1. [5] W. Yang and R.G. Parr, Proc. Natl. Acad. Sci. U.S.A., 82 (1985) 6723. [6] R.G. Pearson, J. Chem. Educ., 64 (1987) 561. [7] R.G. Parr and P.K. Chattaraj, J. Am. Chem. Sot., 113 (1991) 1854. [8] R.G. Parr and J.L. Gbzquez, J. Phys. Chem., 97 (1993) 3939. [9] R.G. Pearson and W.E. Palke, J.Phys. Chem., 96 (1992) 3283. [lo] D. Datta, J. Phys. Chem., 96 (1992) 2409. [l l] S. Hati and D. Datta, J. Org. Chem., 57 (1992) 6056. [12] D. Datta, Inorg. Chem., 31 (1992) 2797. [13] M.K. Harbola, Proc. Natl. Acad. Sci. U.S.A., 89 (1992) 1036. [14] (a) M. Galvbn, A.D. Pino and J.D. Joannopoulos, Phys. Rev. Lett., 70 (1993) 21. (b) A.D. Pino, M. Galvan, T.A. Arias and J.D. Joannopoulos, J. Chem. Phys., 98 (1993) 1606. [15] P.K. Chattaraj, S. Nath and A.B. Sannigrahi, Chem. Phys. Lett., 212 (1993) 223. [16] P.K. Nandi, A. Misra and A.B. Sannigrahi, Proc. Indian Acad. Sci. (Chem. Sci.), in press. [ 171 P.K. Chattaraj and P.v.R. Schleyer, J. Am. Chem. Sot., submitted for publication. [18] G. Lendvay, J. Mol. Struct. (Theochem), 167 (1988) 331. [19] G. Lendvay, J. Phys. Chem., 93 (1989) 4422. [20] D.K. Maity and S.P. Bhattacharyya, J. Am. Chem. Sot., 112 (1990) 3223. [21] (a) K. Fukui, Theory of Orientation and Stereoselection, Springer, Berlin, 1973. (b) K. Fukui, Science, 217 (1982) 747.
309 (1994) 65-77
17
[22] R.G. Parr and W. Yang, J. Am. Chem. Sot., 106 (1984) 4049. [23] J.P. Perdew, R.G. Parr, M. Levy and J.L. Balduz, Phys. Rev. Lett., 49 (1982) 1691. [24] W. Yang and W. Mortier, J. Am. Chem. Sot., 108 (1986) 5708. [25] C. Lee, W. Yang and R.G. Parr, J. Mol. Struct. (Theochem), 163 (1988) 305. [26] F. Mendez, M. Galvan, A. Garritz, A. Vela and J. Gazquez, J. Mol. Struct. (Theochem), 277 (1992) 81. [27] (a) W.J. Hehre, L. Radom, P.v.R. Schleyer and J.A. Pople, Ab initio Molecular Orbital Theory, Wiley, New York, 1986. (b) J.A. Pople and R.K. Nesbet, J. Chem. Phys., 22 (1954) 571. [28] A.B. Sannigrahi and S.D. Peyerimhoff, Chem. Phys. Lett., 112 (1984) 348. [29] A.B. Sannigrahi, T. Kar and P.K. Nandi, Chem. Phys. Lett., 198 (1992) 67. [30] A.B. Sannigrahi, Adv. Quantum Chem., 23 (1992) 301. [31] Z. Latajka and S. Scheiner, J. Chem. Phys., 81 (1984) 4014. [32] F.Y. Fujiwara and J.S. Martin, J. Am. Chem. Sot., 96 (1974) 7625. [33] Z. Zhou and R.G. Parr, J. Am. Chem. Sot., 112 (1990) 5720. [34] (a) C. Parr and H.S. Johnston, J. Am. Chem. Sot., 85 (1963) 2544. (b) H.S. Johnston, Adv. Chem. Phys., 3 (1960) 131. (c) H.S. Johnston, Gas Phase Reaction Rate Theory, Ronald Press, New York, 1966. [35] G.S. Hammond, J. Am. Chem. Sot., 77 (1955) 334. [36] S. Nath, P.K. Nandi, A.B. Sannigrahi and P.K. Chattaraj, J. Mol. Struct. (Theochem), 279 (1993) 207. [37] G. Klopman, in G. Klopman (Ed.), Chemical Reactivity and Reaction Paths, Wiley, New York, 1974, Chapter 4. [38] J. Emsley, Chem. Sot. Rev., 9 (1980) 91. [39] S. Nath, A.B. Sannigrahi and P.K. Chattaraj, J. Mol. Struct. (Theochem), 306 (1994) 87. [40] (a) P.K. Chattaraj and S. Nath, Chem. Phys. Lett., 217 (1994) 342. (b) P.K. Chattaraj and S. Nath, Proc. Indian Acad. Sci. (Chem. Sci.), in press.
Hardness and bond index profiles of hydrogen-bonded complexes with singly-minimum and double-minimum potentials S. Nath, A.B. Sannigrahi, P.K. Chattaraj* Department of Chemistry, Indian Institute
ofTechnology, Kharagpur 721 302, India
(Received 2 November 1993; accepted 11November 1993)
Abstract The dissociation reaction, HsN. * ’ HF + H,N + HF and the proton-transfer reaction (F-H. . * Cl)- -+ fF s. . H-CI)have been studied at the HF//6-31G** level in order to understand the progress of these reactions in terms of global and local reactivity parameters. Both the reactions are found to obey the maximum hardness principle. The potential energy curve for the proton transfer reaction passes through a transition state wherein the hardness is at a minimum. In this reaction bond index profiles of the bonds being broken and formed intersect at the point corresponding to the maximum in the potential energy curve. Condensed Fukui functions have been calculated at different stages of dissociation of H3N. . . HF. Variation of reactivity of a particular site towards electrophilic, nucleophilic or radical attack is properly reflected in these values.
1. Introduction
Understanding chemical reactions is a central theme in chemistry. There have been several attempts to provide theoretical justification of the preference for one reaction path over another and formation of only some selected products among the various possible choices. As early as 1963, based on experimental observations, Pearson [l] proposed that a hard base will prefer to react with a hard acid and a soft base with a soft acid to form a stable product. Pearson’s hard-softacid-base (HSAB) principle has been given a theoretical justification within density functional theory (DFT) [2,3]. In DFT, hardness has been defined as [2]
* Corresponding author.
where p is the electronic chemical potential of an N electron system in the presence of an external potential u(r). The expression for or,is given by [4]
(2) Softness is the reciprocal of hardness [5], i.e. 1 s=% In Eq. (2) E is the total energy of the system. However, within the framework of the BornOppenheimer approximation, the above de& nitions of q and p remain unaltered if electronic energy (Eel) is used in place of E. Using finite difference approximation of Eqs. (1)
016~1280/94/$07.00 0 1994 Elsevier Science B.V. Ail rights reserved SSDI 0166-1280(94)03638-2
S. Nath et al.//. Mol. Struct. (Theochem)
66
and (2) expressions of n and p can be written 12-41 as +%+I
-=,+EN-I
=-
2 -/LU
EN-I
-EN+I
2
_
I-A
2
(4)
Z-i-A
2
(5)
where EN, EN- 1 and EN + I are the energies of N, (N - 1) and (N + 1) electron systems, and f and A are the ionisation potential and electron affinity respectively. The HSAB principle is closely related to the principle of maximum hardness (MHP) which states [6] that “there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible”. The principle of maximum hardness under some constraints has been formally proved by Parr and Chattaraj 171.In a subsequent proof by Parr and Gazquez [8] it was concluded that hardness will be extremum where both electronic energy and nuclear repulsion energy reach their respective extremum values. Computations [9-171 on several molecules, molecular clusters and solids show that higher stability is generally associated with larger hardness values. It has been found at the MNDO level of computations [12] that the exothermic exchange reactions of the type AB + CD + AC + BD would produce the hardest possible species as one of the products. It has also been shown for these reactions that the average hardness of the products is greater than that of the reactants. The same has been found to be true for reactions for which an anomeric effect is operative [ 111,A similar observation was made by Nandi et al. [16] for some proton transfer reactions involving silicon hydrides. However, Chattaraj and Schleyer [ 171found that this is, in general, not true for acid-base reactions leading to binary complexes. They have, however, shown [ 171 through state-of-the-art ab initio calculations that the hardness values of the binary complexes of a given acid with different bases increase in the direction of their increasing complexation energies which in most cases match with the direction predicted by the HSAB principle. Hardness profiles for the inversion of NH3 [lo] and PH3 [15] and for intramolecular hydrogen transfer reaction of malonaldehyde [lo] which involves large acti-
309 (1994) 65-77
vation barriers are found to pass through a transition state where hardness is at a minimum and energy is at a maximum. The course of a reaction can also be followed using charge density-related parameters like bond index etc. It seems “that there exists a parallelism between bond index and charge rearrangement in the course of a reaction” [18]. The bond index of the bond being broken or formed in a reaction passes through an inflection point at which the bond index profiles intersect [l&20]. This inflection point closely corresponds to the transition state predicted from the energy profile provided there is a similarity between the bonds being broken and formed. It is known that when a reactant molecule is approached by another, the frontier orbitals play the most significant role since charge distribution in these orbitals determines the stereoseiective behaviour of a reaction [21]. In DFT, the reactivity at different sites of a chemical species can be predicted from local parameters like local softness (S(Y)) [5] which is proportional to the Fukui function (S(u)) [22].
where f
(r)
is defined [22] as
Since in atomic and molecular systems E vs. N plots are discontinuous [23] three different Fukui functions can be obtained by finite difference approximation of Eq. (7) for electrophilic, nucleophilic and radical attack 122-251. Useful information about stereoselective attacks can also be obtained from analysis of the variation of atomic charge in molecules which in turn can be employed to define [24,25] condensed Fukui functions. The condensed Fukui functions have been shown to account for the gas phase basicity of amines, and electrophilic and nucleophilic attacks on several molecules and ions [24-261. In the present investigation we have studied using the ab initio SCF method 1271the course of chemical reactions involving two hydrogen-bonded
61
S. Nath et al.lJ. Mol. Struct. (Theochem) 309 (1994) 65-77
species, namely H,N. . . HF and (FHCI)-. The former one is a normal hydrogen-bonded system with a single minimum potential and the latter is a very strongly hydrogen-bonded system characterised [28,29] by a double-minimum potential corresponding to the conformers (F-H.. . Cl)- and (F ... H-Cl)-. Thus, we have confined our attention to the following reactions. HsN...HF
+ H,N+HF
(F-H..eCl)-
+ (F..&H-Cl)-
(1&, vaiency ( VA) and molecular valency ( VM) are defined [30] by
qA=&aa
(12)
a
(13)
(8)
(14)
(9)
(15)
Reaction (8) leading to the dissociation products takes place along the Cs axis passing through the N. . . H -F moeity. Along this coordinate the reaction does not pass through any transition state which may, however, appear along some other reaction coordinate in the energy hypersurface. In contrast, reaction (9) is associated with a very high energy barrier with (F..-H .+ ‘Cl)- being the transition state (TS). molecular The hardness, chemical potential, valency and bond index profiles have been calculated over the entire course of reactions (8) and (9) to investigate the progress of the reaction. Alternative definitions of n and ~1have been compared. Condensed Fukui functions are calculated at different steps of the NH, . . . HF dissociation to predict sites for the electrophilic, nucleophilic or radical attack. Finally, the reactions have been discussed in the light of the MHP.
where D is the first order density matrix whose explicit form depends upon the scheme of population analysis used. We have used here the commonly used schemes based on Mulliken’s and Lowdin’s population analyses abbreviated as MPA and LPA respectively. The respective density matrices are given by DMvIpA = PS
(16)
DL~A = S”2PS”2
(17)
where P = 2Cf? (C is the coefficient matrix of the doubly occupied MOs) and S is the A0 overlap matrix. The condensed Fukui functionsfkf (for nucleophilic attack), f; (for electrophilic attack) and f{ (for radical attack) are calculated using the following expressions. (18)
2. Method of calculation Ass~ing the validity of Koopmans’ Eqs. (4) and (5) can be written as {2-41 n=
(fLUM0
-
F =
(‘~0~0
+ ~LUMO)/~
EHOM0)/2
.fk
theorem
(10)
(11)
where E denotes orbital energy. In view of their simplicity, Eqs. (10) and (11) have been used for the calculations of n and p. These quantities have also been calcuIated in a few cases using Eqs. (4) and (5). The gross atomic charge (qA), bond index
= qk(N)
-
qk(N
fko=lf21qk(N+1)-qktN-t)j
-
1)
(19) (20)
Here qk is the gross charge on atom k in molecules having N, N + 1, N - 1 electrons. All calculations have been carried out at the HF//6-31G** level. The geometrical parameters for NHs, HF and H3N.. . HF have been taken from literature [31]. Along the dissociation path only the N . . . H distance is varied keeping r(NH) and 8 < HNH fixed at their monomer value and r(H-F) optimised for each value of r(N .+- H). Calculations on (FHCl)- have been performed at r(F ..+Cl) = 3.0, 4.0 and 5.0A. At
68
S. Nath et al./J. Mol. Struct. (Theo&em) 309 (1994) 65-77 -2.60
910
-3.60 6.80
- -023s
I
14Kl
2.00
_3.00
k..
1
600 500 r ( N----H) CA) --c
I
6ffl
Fig. 1. Variation of total energy (E) (----_X hardness fq) I- - -). chemical potential &) (- x - x --I$ molecular valency (FM) c s.f, and bond index I- + - m-f of the H$J.. I HF molecule along the reaction coordinate r(N ~. . H).
each distance the position of H is varied to get the potential energy curve. In order to locate the actual transition state of reaction (91, several r(F +ef Cl) values should have been considered_ The chosen distances are, however sufficiently representative to mimic the nature of the double-minimum potential. In order to determine the condensed Fukui functions and to compare the performance of Eqs. (4) and (5) in the calculation of n and p* the energy and atomic charge of (HsN I ) ~RF)+ and (I-IsN . * . HF)- have been calculated using a UHF formalism [27],
Figure 1 depicts the variation of total energy, hardness, chemical potential, I&, and IN_ _n of H3N. . . HF along the reaction coordinate, r(N - -H), As expected, the energy profile shows a minimum at the equilibrium geometry
of the complex and then rises sharply to reach the dissociation limit, which has been taken here to be r(N - -H) = 6 A. At this distance the binding energy of the complex is only -0.53 kcal mol-’ . The calculated hydrogen-bond energy (- 11.82 kcal mall’) compares favourably with the literature value of - 13.14 kcalmol-’ calculated at the MP4SDTQ level with the 6-3 115 G** basis set and the corresponding ZPE corrected value of -10.35 kcal mol-’ [17’f. The hardness profile along the dissociation path shows that r] is not the maximum at the equilibrium distance corresponding to the lowest value of the energy, as is apparently demanded by MHP. Further, rf passes through a shallow minimum around r(N - -H) $=3 A, although there is no maximum in the energy profile around that point, Both of these features can be understood if the role of the nuclear repulsion potential (V,,) is considered in finding out the conformation with maximum hardness. Although not shown here, the
S. Nath et al./J. Mol. Struct.
(Theochem)
309 (1994) 65-77
69
Table 1 Total energy (_&I, a.u.), hardness (7, eV), relative hardness (nr+ eV), chemical potential (p, eV), relative chemical potential (~~1, eV), molecular valency ( VM), relative molecular valency ( VM)reI and bond index (I,...,) of the N. . H bond along the reaction coordinate rN...n (A) rN...u
-E
7”
-Pa
VM
IN-H
%I b
-A.4
1.6
156.2298
9.021
3.868
3.517
0.083
1.806c
156.2257
8.999
3.797
3.543
0.072
2.206
156.2227
8.873
3.609
3.574
0.057
2.406
156.2203
8.811
3.517
3.580
0.062
3.0
156.2148
8.712
3.269
3.591
0.021
4.0
156.2099
8.734
2.980
3.589
0.001
5.0
156.2083
8.728
2.854
3.595
0.0
6.0
156.2077
8.719
2.800
3.597
0.0
0.301 0.315 0.280 0.293 0.154 0.167 0.092 0.105 -0.007 0.006 0.015 0.028 0.009 0.022 0.0 0.013
1.068 1.203 0.997 1.131 0.809 0.943 0.717 0.852 0.469 0.604 0.180 0.315 0.054 0.188 0.0 0.134
b
( VM )rel b
-0.080 0.782 -0.054 0.808 -0.023 0.839 -0.017 0.844 -0.006 0.855 -0.008 0.854 -0.002 0.860 0.0 0.862
a Hardness (9) and chemical potential (h) have been calculated using Eqs. (6) and (7) respectively. b The first set of numbers denotes values relative to those of HsN t. HF at r(N H) = 6A and the second set of numbers denotes values relative to those of free NHs (n = 8.706eV, p = -5.331 eV, VM = 2.736). ’ Equilibrium value.
electronic energy (E,,) increases monotonically along the dissociation path and the nuclear repulsion decreases in a somewhat similar manner. Neither of them shows any extremum along the reaction coordinate which is a necessary condition for n to attain an extremum value [8]. The opposite trend in the variations of Z&i and V,, causes the hardness profile to pass through a shallow minimum. It is worthwhile mentioning in this context that the importance of V,,, in locating the maximum hardness conformation, has been discussed earlier in the cases of symmetric and asymmetric vibrations of molecules [9,15]. It has been found that in the latter mode V,, remains virtually constant and hardness is a maximum at the equilibrium geometry where Z&t is a minimum. However, in symmetric vibrations, V,, does not remain constant and neither Z&tnor r] becomes an extremum at the equilibrium state. The chemical potential (p) increases and tends to attain a constant value at the dissociation limit (Table 1). The fact that peues < z+,N...HF), indicates that the complex at the equilibrium geometry
has a greater tendency to accept electrons than at the infinitely separated monomer limit. The increase in molecular valency ( VM) is ve:y sharp at the beginning and from r(N . . . H) M 3 A onwards the change becomes rather negligible. Along the dissociation path the change in VM is due to changes in ZN.,,u and ZnF. The former decreases and the latter increases with increasing r(N . . . H). Upto r(N.. . H) x 3 A, the increase in In-F outweighs the decrease in ZN,,,H and Vj increases. Beyond this point ZN...n is negligible and ZH_F remains virtually constant. The variations of VM and n relative to that of NH3 and of (HsN . . . HF)oo (not shown in the figures) follow the same pattern and can be explained using the above arguments. The nature of the variations of n relative to those of NH3 and (HsN... HF)oo implies that n(complex) > r](complex) > n(HsN . . . HF),, rl(NHs), and which are in accordance with the HSAB theory and MHP (Table 1). The bond index profile (Fig. 1) of the N-H bond shows that it is somewhat erratic around
S. Nath et al./J. Mol. Struct.
70
(Theochem)
309 (1994) 65-77
Table 2 Variation oftotal energy (E, a.u.), hardness (q, eV), chemical potential (p, eV), molecular and H-Cl (ZH_& bonds with increasing F-H distances @(F-H), A)”
valency
( VM),and bond indices of F-H (&,)
r(F-H)
-E
9
0.8 0.8 0.8
559.5479 559.5389 559.5309
9.597 7.768 6.994
5.320 3.854 3.325
0.981 0.938 1.195
0.878 0.906 0.917
0.108 0.034 0.004
0.947 0.947 0.947
559.5706 559.5578 559.5477
9.498 7.435 6.555
4.929 3.387 2.822
0.877 0.841 0.824
0.727 0.797 0.819
0.146 0.044 0.005
1.0 I.0 1.0
559.5668 559.5519 559.5408
9.437 7.275 6.355
4.735 3.854 2.596
0.854 0.819 0.801
0.681 0.768 0.795
0.165 0.049 0.006
1.2 1.2 1.2
559.5368 559.5087 559.4929
9.263 6.606 5.502
3.912 2.222 1.633
0.838 0.791 0.769
0.540 0.703 0.757
0.270 0.079 0.010
1.5 1.5 1.5
559.5041 559.4407 559.4115
9.216 5.952 4.383
3.510 0.865 0.264
0.920 0.843 0.791
0.372 0.631 0.754
0.519 0.176 0.030
1.7 1.7 1.7
559.4907 559.4109 559.3665
9.000 5.904 3.931
4.239 0.220 -0.466
0.978 0.891 0.818
0.296 0.549 0.732
0.683 0.284 0.067
2.0 2.0 2.0
559.3825 559.3958 559.3189
8.894 5.802 3.738
5.167 0.217 - 1.260
1.042 0.931 0.880
0.251 0.386 0.637
0.853 0.478 0.189
2.7 2.7 2.7
551.6759 559.4472 559.3004
7.287 6.407 3.494
4.590 3.245 - 1.027
0.887 0.942 0.926
-0.017 0.097 0.591
0.826 0.847 0.263
’ Three values in each column
correspond
to fixed F.
P
Cl distances
r(N . . H) x 2A and seems to pass through an inflection point around r(N . . H) M 3 A which may be reminiscent of the minimum n value at this point. Some representative values of various quantities calculated for (FHCl)) are given in Table 2. The potential energy (PE) curves of this species resulting from the variation of energy with respect to r(F-H) are shown in Fig. 2 for r(F . . . Cl) = 3, 4 Dand 5 A. The PE curve at r(F.. . Cl) = 3 A (it is less than the sum (3.15 A) of the van der Waals radii of F and Cl) passe: through a single minimum. At r(F . . . Cl) > 3.15 A, i.e. at 4.0 and 5.OA the PE curves exhibit the doubleminimum characteristics. In the last cases the TS, (F..O. H. . . Cl)- corresponds to r(F-H)=2.0 A and 2.5 A respectively. ’
.
VM
IF-H
[H-Cl
of 3 A, 4 A and 5 A respectively.
The hardness profiles (Fig. 3) at r(F . . . Cl) = 4 A and 5A pass through a broad minimum in each case corresponding to the TS. The positions of the minima are however not exactly identical to those of the maxima in the energy profiles since the positions of respective extremum in E,i and I’,, profiles (not shown here) do not coincide. As in the case of HsN . . . HF, the hardness is not a maximum at the equilibrium geometry owing to well-known reasons. However, the n value corresponding to the minimum (9.498eV) in the PE curve at r(F.. . Cl) = 3 A is considerably greater than the respective value at longeroF. . . Cl distances (7.275 eV at r.(F . . . Cl) = 4A, and 6.555eV at r(F.. . Cl) = 5 A). It has been noted earlier [28] that the energy of activation required for reaction (9) is less than that for the reverse
S. Nath et al./J. Mol. Struct.
(Theochem)
71
309 (1994) 65-77
- 0.00 I
I
I
-0.10
I
I I
I
-02c l-
t
I
c ?
a
=: -0.3c z ul + w -
/ \
-Odd
\
I
/’
-05
-0% O0.00
Fig. 2. Variation
of total energy
I
I
I
I
2Gl
3.00
L.00
500
r
( F-H I(%) --c
I
1.00
(E) of (FHCl)-
with r(F-H)
reaction which agrees with the experimental observation [32] that in the ion-molecule exchange reaction, F- + HCl = HF + Cl-, the proton is transferred to the halide ion with lower atomic number. The hardness profile of this reaction shows that r](F-H . . . Cl)- decreases by a greater margin than n(F . . . H-Cl)while forming the activation complex (F-..H . ..Cl)- with minimum hardness. It follows from the MHP that a system in equilibrium with maximum hardness resists any tendency of lowering in hardness value and consequently the process of forming the activation complex from (F.. . H-Cl)will be easier than that from (F-H. . . Cl)-. In agreement with this, both the hardness profile and the PE curve show the preference of the backward reaction to occur. It can also be noticed from Fig. 2 that the energy barrier is higher at r(F . . . Cl) = 5 A than at 4A. This tallies with the observation that the hardness profile corresponding to the formation of the activation complex is deeper at r(F . . .Cl) = 5A than at r(F...Cl) =4A. It again follows from the MHP that the reaction
at r(F
Cl): 3 A, (-
. - . -); 4& (-),
5 A (-
-
-),
path which has a higher activation barrier and in consequence lesser reactivity will form a TS complex having a lower hardness value. It may be mentioned in this context that the preference of reaction path can also be explained in terms of activation hardness defined by Zhou and Parr [33]. The chemical potential for the system is positive over the entire region for r(F . . . Cl) = 3 and 4 A (Fig. 4) and attains a small negative value at the transition state for r(F . .. Cl) = 5 A. The positive p values imply that (FHCl)-, already a stable closedshell anion, does not have any tendency to accept electrons. However, the TS at r(F.. . Cl) = 5 A seems to behave differently. The Vhl profile (Fig. 5) at r(F es. Cl) = 3 A shows the pre!ence of0 one deep minimum. At r(F . . . Cl) = 4 A and 5 A the presence of another minimum of smaller depth can be noticed. However, the overall variation of VM in the process is rather small. The minimum value of Vhl near the equilibrium geometry emphasises the importance of electrostatic interaction in the formation of (FHCl)-. Moreover, it is interesting to note from
72
S. Nath et al/J.
-\ 7.00
\ \
Mol. Struct. (Theochem)
\
\\ \\
t q(QV)
\
309 (1994) 65-77
/ P /I
J
! : I
\ I
\
ml
\
:
\ \
3oc i
0
I \
/
\ __--
\
\_ ,I’
I
I
2.00
300
r(F-H)(I%) Fig. 3. Variation
of hardness
(7) of (FHCI)-
with r(F-H)
‘\ 4.00
\\
\\
\
!
t
Cl):
3 A, (- .
- . -); 4 A, (-),
!
/
‘\.J
/I
5A (-
- -).
/
I /
i
\ \
I
\ \
1
$eV
-
at r(F.
\ 2.00I-
500
&OO
!./‘-
\
I
I
I
1.00
\ \
\
\
o*ot)\ \
- 2.0( l-
IO o-a
I
l-00
\
/
/
/
1-l’ I
I
2.00
390
I
I
LB.00
500
r (F-H)(%)Fig. 4. Variation
of chemical
potential
(p) of (FHCl)-
with r(F-H)
at
r(F
Cl):
3 A, (-
* - . -); 4A, (-),
5 A (-
- -).
S. Nath et al./J. Mol. Struct.
lo
Fig. 5. Variation
of molecular
valency
(Theochem)
1
I
l%O
200 r(F-H)( ii) -
( VM) of (FHCl)-
I
with r(F-H)
the V, and p profiles at r(F . . . Cl) = 4 A and 5 A that these quantities reach their respective maximum and minimum values around the TS geometry. The variation of bond indices of the bonds being broken (F-H) and formed (H-Cl) along the reaction coordinate has been presented in Fig. 6. The nature of the curves is almost identical for all three F. . . Cl distances. At the F. +. Cl distances of 4 A and 5A, the Zr_, and Zu_cr curves intersect at a point which corresponds (within 0.05-o. 1 A) to the maximum in the respective potential energy curves. This agrees with earlier SCF and INDO/MCSCF calculations [18-201 on systems where the bonds being broken and formed are similar. It has been observed that for thermoneutral reactions the two bond indices are equal at TS in accordance with the principle of conservation of bond indices [34] and Hammond’s principle [35]. However, for exothermic and endothermic reactions these bond indices should not be the same at the saddle point. This has been found to be valid for several types of exchange reactions studied by Lendvay [ 18,191. In the work of Maity and Bhattacharyya [20] inflec-
13
309 (1994) 65-77
I
3.ocl
at r(F
Cl):
&OO
3 A, (- .
1 500
- . -); 4A, (-),
5 A (-
- -).
tion points in the bond index profiles of N-H and O-H bonds in the isomerisation of HNO to NOH are found to coincide with the actual saddle point of the reaction as obtained from the corresponding PE curve. Since this reaction has been found to be an endoergic one, a greater value of ZNHthan ZOH (as is found from Table I of Ref. 20) at the TS seems to be against Hammond’s postulate. We would like to clarify that the intersection point of two bond index profiles around respective inflection points should not be used to locate the saddle point in reactions other than thermoneutral ones because the bond indices will have same value at the TS only for thermoneutral reactions according to Hammond’s principle. It is however, gratifying that in the presently studied reaction (9) of (FHCl)-, which is also an endoergic one, the bond index values at TS show that ZHcl is slightly greater than ZFH (Table 2) which is in accordance with Hammond’s principle [35]. It is interesting to note that the ,,minima in hardness profiles at r(F . . . Cl) = 4A and 5 A (Fig. 3) lie towards the product side which may also be thought of as a consequence of Hammond’s principle.
14
S. Nath et al./J. Mol. Strucr. (Theochem)
r(F-HI
Cli,
309 (1994) 65-77
-
Fig. 6. Variation of bond indices of bonds being broken (I,_,+) .. and formed (ZfI_cI) of (FHCI)- with r(F-H) at r(F (- * - ‘ -); 4‘4, (-), 5A (- - -). -
The condensed Fukui functions calculated for I&N. s’ HF at its equilibrium geometry, dissociation limit and an intermediate point, have been presented in Table 3. Energies of neutral HsF . v- HF and the corresponding positively and negatively charged species along with n and ,LL values calculated from definitions (4) and (5) (using vertical I and A values) have also been reported. The relative values of 7 and ,u obtained using these equations follow the same order, although the numerical values differ owing to the iimitations ofthe Koopmans’ theorem which has the well-known tendency to overestimates and A values. The three sets of condensed Fukui functions revea1 that the value off- is largest for nitrogen which means that due to the presence of a lone pair of electrons it will be the most favourable site for electrophilic attack. The hydrogens attached to nitrogen have a high f+ value and should be potent centres for nucleophilic attack. This agrees with the observed high dipole moment of NHs. The value of ff is very small for hydrogen-bonded hydrogen since some of the lone pair density of
. Cl): 3A,
nitrogen has been transferred to hydrogen due to the inductive effect of fluorine. Nitrogen shows a greater tendency to bind electrophilic reagents (fis higher) than fluorine. In the course of the dissociation reaction, as HF moves away from NH3, condensed Fukui functions show expected trends. The propensity for electrophilic attack on nitrogen increases as r(N . a. H) increases because the lone pair of electrons becomes relatively free. The lower f+ value of the three hydrogens attached to nitrogen and the higher f’ value of the hydrogen engaged in three-centre hydrogen-bonding interaction at r(N sI -H) = 2.206A is in confo~ity with the classical chemical concept. As the hydrogen-bonded hydrogen moves away from the lone pair of nitrogen, it becomes a better nucleophilic centre and hydrogens attached to nitrogen become less prone to nucleophilic attack. Howeve:, near the dissociation limit, at r(N. . . H) = 6.0 A, this trend of the f' of hydrogens gets reversed which may be due to the localisation of negative charge on NH3 in the [NH,. . . HF]- species as predicted by the population analyses (MPA and LPA).
S. Nath et al./J. Mol. Struct.
(Theochem)
15
309 (1994) 65.77
Table 3 Energies (E, au.) of (H3N. . HF)+, HsN. HF and (HsN I HF)- species, hardness (n, eV), and chemical potential (p, eV), and condensed Fukui functions for the HsN. . HF molecule along the reaction coordinate r(N H) Atom r(N.
-Ea
li”
+C
Pb
fk
7.752
2.857
N
H’d H” d F .H)
7.620
0.4911 0.6068 0.1447 0.0952 -0.0577 4.47 x 10-r 0.1326 0.1030
-0.4138 0.1322 0.4410 0.2641 0.0549 0.0419 0.0364 0.0341
0.5116 0.6581 0.1494 0.0972 -0.0495 -0.0118 0.0898 0.0618
-0.3948 0.1601 0.4648 0.2799 -9.04 x 10-s -5.52 x lo-’ 9.05 x 1o-3 5.52 x 1O-3
0.5216 0.6932 0.1595 0.1022 -0.0118 -7.14 x 10-s 0.0118 7.14 x 10-s
0.0293 0.3687 0.2967 0.1814 -6.60 x 10-r 0.0188 0.0845 0.0686
2.641
N
H’ H” F .H)
-0.4324 0.1307 0.4487 0.2676 0.0511 0.0331 0.0364 0.0342 =2.206/i 155.8456 156.2227 156.0397
r(N.
fi”
H) = 1.806d 155.8358 156.2257 156.0458
r(N.
fiC
0.0489 0.3951 0.2952 0.1806 2.70 x lO-3 0.0150 0.063 1 0.0479
=6.0/i 155.8630 156.2083 156.0021
7.496
-1.840
N
H’ H” F
0.0634 0.4266 0.3121 0.1910 -0.0142 -6.33 x IO-’ 0.0142 6.33 x 1O-3
a Three values correspond to the energy of (H3N.. HF)+, H3N.. HF and (H3N . HF)- species respectively. b n and p values are calculated from Eqs. (4) and (5) respectively. ‘The first and second values off:, f ;, f e have been calculated by using MPA and LPA gross charges respectively. d H’ denotes hydrogen attached to nitrogen in NH3 molecule and H” denotes hydrogen-bonded hydrogen in H3N. . HF.
The reactivity trend of the molecule along the dissociation path as obtained from the study of condensed Fukui functions remains the same if either of the MPA and LPA schemes is used. Note that the radical attack on N and H’ centres are, however, predicted differently in these population analysis schemes. The numerical values of
Fukui functions obtained from LPA are lower than those obtained using MPA since LPA overestimates covalent interaction [36]. This overestimation leads to an unusual result in the case of radical attack. While in MPA scheme the radical attack is favoured at three hydrogen atoms, the nitrogen atom is preferred in the LPA scheme. In
76
S. Nath et a1.l.J.Mol. Struct. (Theochem) 309 (1994) 65-77
general, anywhere along the dissociation path f &, fi and_& (f& in LPA scheme) possess the largest values. Since the Fukui function is a normalised local softness (Eq. (6)) a soft nucleophile or a soft radical will prefer to attack at the hydrogen atom of NH3 and a soft electrophile will prefer to bind to the nitrogen atom. Since the hard-hard interaction is supposed to be charge controlled [37] as opposed to the frontier-controlled soft-soft interactions and the total gross charge is the largest (for all f(N . . . H) distances and for both LPA and MPA schemes) on the nitrogen atom, any hard species or hard end of a species would prefer to attack the nitrogen site of the H,N . . . HF. Overall consideration of the two reactions shows that both of them obey the MHP. The formation of HsN.. . HF is exoergic to the extent of 11.82 kcal mol-’ and is accompanied by an increase in n of 0.28eV. In general n decreases from its equilibrium value along the dissociation path which is accompanied by an increase in energy as well as chemical potential. Note that both definitions for n and p (Eqs. (4) (5), (10) and (11)) provide identical trends. Thus, the MHP has been found to be valid although p is not strictly a constant which was a constraint in the Parr-Chattaraj proof [7]. The energy profile of the (FHCl)) ion shows that the minima can be approached from the TS (F...H . ..Cl)- in two ways. Ab initio calculations [28] as well as experimental results [38] show that the formation of the species in which fluorine is closer to hydrogen will be the favourable process since [F-H . . . Cl]- is associated with the minimum energy and the maximum hardness [7]. This is illustrated below u$ng E and n values from Table 2 for r(F...Cl) = 5A. [F...H-Cl]-
[F . . . H . . . Cl]-
going from one species to the other hardness decreases and vice versa which is in conformity with the MHP. Chattaraj and co-workers have noticed the validity of the MHP in various static [ 15,17,39] and dynamic [40] situations.
4. Conclusion This work presents an analysis of chemical reactions associated with single and double minimum potentials in terms of the HSAB theory and MHP. For reactions passing through a TS, the hardness profile is found to pass through a minimum. The greater well depth in the hardness profile is related to a lesser reactivity as has been the case with a greater activation energy barrier. The nuclear repulsion potential is very important in locating the state with maximum or minimum hardness. Hammond’s principle has been discussed from energy, bond index and hardness profiles. The study of condensed Fukui functions for HsN . . . HF dissociation gives general information about gradual accumulation or depletion of charges on atoms in a molecule which in turn determines the centre for the nucleophilic or electrophilic attack in a chemical reaction. Comparison of hardness values of reactants and products show that reactants proceed in a direction in which the product with maximum hardness is formed which is in agreement with the MHP.
5. Acknowledgement
PKC would like to thank the Council of Scientific and Industrial Research (CSIR) for financial help.
E - 559.408 a.u. + -559.293 a.u.
n
4.45eV
3.74eV + [F-H.
. . Cl]-
-559.548 a.u. 6.55 eV Comparison of n values of different species in the above reaction shows that if energy increases in
6. References [l] (a) R.G. Pearson, J. Am. Chem. Sot., 85 (1963) 3533. (b) R.G. Pearson (Ed.), Hard and Soft Acids and Bases, Hutchinson and Ross, Stroudsburg, PA, 1973. [2] (a) R.G. Parr and R.G. Pearson, J. Am. Chem. SOL, 105 (1983) 7512.
S. Nath et al./J. Mol. Struct. (Theochem)
(b) P.K. Chattaraj, H. Lee and R.G. Parr, J. Am. Chem. Sot., 113 (1991) 1855. [3] (a) R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. (b) P.K. Chattaraj and R.G. Parr, in K.D. Sen and D.M.P. Mingos (Eds.), Chemical Hardness, Springer, Berlin, 1992, p. 11. (c) P.K. Chattaraj, J. Indian Chem. Sot., 69 (1992) 173. (d) M.K. Harbola, P.K. Chattaraj and R.G. Parr, Isr. J. Chem., 31 (1991) 395. [4] R.G. Parr, R.A. Donnelly, M. Levy and W.E. Palke, J. Chem. Phys., 69 (1978) 443 1. [5] W. Yang and R.G. Parr, Proc. Natl. Acad. Sci. U.S.A., 82 (1985) 6723. [6] R.G. Pearson, J. Chem. Educ., 64 (1987) 561. [7] R.G. Parr and P.K. Chattaraj, J. Am. Chem. Sot., 113 (1991) 1854. [8] R.G. Parr and J.L. Gbzquez, J. Phys. Chem., 97 (1993) 3939. [9] R.G. Pearson and W.E. Palke, J.Phys. Chem., 96 (1992) 3283. [lo] D. Datta, J. Phys. Chem., 96 (1992) 2409. [l l] S. Hati and D. Datta, J. Org. Chem., 57 (1992) 6056. [12] D. Datta, Inorg. Chem., 31 (1992) 2797. [13] M.K. Harbola, Proc. Natl. Acad. Sci. U.S.A., 89 (1992) 1036. [14] (a) M. Galvbn, A.D. Pino and J.D. Joannopoulos, Phys. Rev. Lett., 70 (1993) 21. (b) A.D. Pino, M. Galvan, T.A. Arias and J.D. Joannopoulos, J. Chem. Phys., 98 (1993) 1606. [15] P.K. Chattaraj, S. Nath and A.B. Sannigrahi, Chem. Phys. Lett., 212 (1993) 223. [16] P.K. Nandi, A. Misra and A.B. Sannigrahi, Proc. Indian Acad. Sci. (Chem. Sci.), in press. [ 171 P.K. Chattaraj and P.v.R. Schleyer, J. Am. Chem. Sot., submitted for publication. [18] G. Lendvay, J. Mol. Struct. (Theochem), 167 (1988) 331. [19] G. Lendvay, J. Phys. Chem., 93 (1989) 4422. [20] D.K. Maity and S.P. Bhattacharyya, J. Am. Chem. Sot., 112 (1990) 3223. [21] (a) K. Fukui, Theory of Orientation and Stereoselection, Springer, Berlin, 1973. (b) K. Fukui, Science, 217 (1982) 747.
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[22] R.G. Parr and W. Yang, J. Am. Chem. Sot., 106 (1984) 4049. [23] J.P. Perdew, R.G. Parr, M. Levy and J.L. Balduz, Phys. Rev. Lett., 49 (1982) 1691. [24] W. Yang and W. Mortier, J. Am. Chem. Sot., 108 (1986) 5708. [25] C. Lee, W. Yang and R.G. Parr, J. Mol. Struct. (Theochem), 163 (1988) 305. [26] F. Mendez, M. Galvan, A. Garritz, A. Vela and J. Gazquez, J. Mol. Struct. (Theochem), 277 (1992) 81. [27] (a) W.J. Hehre, L. Radom, P.v.R. Schleyer and J.A. Pople, Ab initio Molecular Orbital Theory, Wiley, New York, 1986. (b) J.A. Pople and R.K. Nesbet, J. Chem. Phys., 22 (1954) 571. [28] A.B. Sannigrahi and S.D. Peyerimhoff, Chem. Phys. Lett., 112 (1984) 348. [29] A.B. Sannigrahi, T. Kar and P.K. Nandi, Chem. Phys. Lett., 198 (1992) 67. [30] A.B. Sannigrahi, Adv. Quantum Chem., 23 (1992) 301. [31] Z. Latajka and S. Scheiner, J. Chem. Phys., 81 (1984) 4014. [32] F.Y. Fujiwara and J.S. Martin, J. Am. Chem. Sot., 96 (1974) 7625. [33] Z. Zhou and R.G. Parr, J. Am. Chem. Sot., 112 (1990) 5720. [34] (a) C. Parr and H.S. Johnston, J. Am. Chem. Sot., 85 (1963) 2544. (b) H.S. Johnston, Adv. Chem. Phys., 3 (1960) 131. (c) H.S. Johnston, Gas Phase Reaction Rate Theory, Ronald Press, New York, 1966. [35] G.S. Hammond, J. Am. Chem. Sot., 77 (1955) 334. [36] S. Nath, P.K. Nandi, A.B. Sannigrahi and P.K. Chattaraj, J. Mol. Struct. (Theochem), 279 (1993) 207. [37] G. Klopman, in G. Klopman (Ed.), Chemical Reactivity and Reaction Paths, Wiley, New York, 1974, Chapter 4. [38] J. Emsley, Chem. Sot. Rev., 9 (1980) 91. [39] S. Nath, A.B. Sannigrahi and P.K. Chattaraj, J. Mol. Struct. (Theochem), 306 (1994) 87. [40] (a) P.K. Chattaraj and S. Nath, Chem. Phys. Lett., 217 (1994) 342. (b) P.K. Chattaraj and S. Nath, Proc. Indian Acad. Sci. (Chem. Sci.), in press.