8121
J. Phys. Chem. 1995,99, 8121-8124
Hardness Profiles of Some 1,2-Hydrogen Shift Reactions Tapas Kar and Steve Scheiner* Department of Chemistry, Southem Illinois University, Carbondale, Illinois 62901 Received: December 22. 1994@
Ab initio SCF/6-31G** calculations are performed on the 1,Zhydrogen shift reaction in some HAB molecules and HAB+ ions (AB = CN, SiN, BO, A10, BS, AlS, BeF, CO, SiO, CS, N2) in order to test the validity of the maximum hardness principle. The hardness is found to be a good indicator of the more stable isomer in most cases, as is the chemical potential. Profiles of these quantities, as a function of the reaction coordinate, do not exhibit extrema at the transition state, but other extrema are noted at unexpected points. While the total energy does not exhibit the same behavior as the hardness or the chemical potential, the individual electronic and nuclear repulsion energies are in close parallel with them.
Introduction The maximum hardness principle (MHP) for atomic and molecular systems has become an active field of research in recent years. This important principle of molecular electronic structure theory was originally introduced by Pearson:’ “there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible”. The hardness (4) of a system is related to the second derivative of energy with respect to the number of electrons N , for constant potential, due to nuclei plus extemal potential (v) and temperature ( r ) . The slope of this quantity under these conditions is the chemical potential p. According to the MH principle, the system will evolve to a state with maximum q at constant T, v, and p . Parr and Chattaraj2 recently provided a formal proof, using the fluctuationdissipation theorem of statistical mechanics. Hardness has been conceived as a measure of the stability of a system. Numerical results3-I3 which support the MHP have started accumulating. All these studies indicate that the principle is potentially a very powerful tool for the study of molecular electronic structure. Most of the previous calculations reported earlier on the validity of the MHP concerned molecular deformations and internal rotations. These studies indicate that the hardness profile contains a minimum for reactions passing through a transition state, even though there is no rigorous reason why this must be so. It was intriguing to note from a recent study of the HCP-HPC isomerization reactionI3 that the hardness profile passes through a minimum where the transition state might be expected to occur. However, HPC is not a minimum in the ground state potential surface,I4 and the potential energy surface (PES) does not pass through a maximum where the hardness indicates this transition state. The purpose of the present investigation is to examine the validity of the MHP over a range of isomerization reactions. For this purpose we have considered the 1,Zhydrogen shift in some HAB (AB = CN, SiN, BO, A10, BS, AlS, BeF) molecules and HAB+ (AB = CO, SiO, CS, N2) ions. The energetics of some of these reactions have been studied previou~ly;’~ they are associated with a range of barrier heights separating the two minima on their PES.
Method of Calculation The hardness and chemical potential are rigorously defined asI6 @
Abstract published in Advance ACS Abstracts, April 15, 1995.
0022-3654/95/2099-8121$09.00/0
where v is the extemal potential and N the number of electrons. Using a finite difference approximation, it can be shown that”
17 = (IP - EA)/2 ,U = -(IP
+ EA)/2
(3)
(4)
where IP and EA refer to the first ionization potential and first electron affinity, respectively, of the N-electron system. Assuming the validity of Koopman’s theorem, one can write’
where E denotes the energy of the lowest unoccupied or highest occupied molecular orbital. All calculations have been performed at the SCF level using the polarized split-valence 6-31G** basis set. It has been shown by Nath et a1.I0 that such a medium-sized set, which is satisfactory for other properties, can be used also for successful calculation of 17. After locating the transition state (TS) geometries of the isomerization reactions, the minimum energy path (MEP) was followed in both directions (forward and reverse) using the intrinsic reaction coordinate (IRC) method.’* Finally, 17 and p were calculated at each point of the MEP using eqs 5 and 6. We have also computed two- (IAB)and threecenter (I& bond indices and molecular valency (VM) using the standard methodIg in order to verify the maximum molecular valency principle (MMVP).” According to MMVP, the molecular valency of a molecular should be maximized at its equilibrium geometry.
Results and Discussion Transition state (TS) and equilibrium geometries of the isomers, HAB and HBA, were fully optimized at the SCF level using the 6-31** basis set. While some of the calculated isomerization energies (hE) are in good agreement with correlated MP3 result^,'^ others differ significantly. It is worth stressing that we are interested here not so much in the accuracy of the isomerization energies and the barrier heights but rather 0 1995 American Chemical Society
8122 J. Phys. Chem., Vol. 99, No. 20, 1995
Kar and Scheiner
TABLE 1: Total Energy (E), Hardness ( p ) , Chemical Potential @), and Molecular Valency ( V M )for HAB Molecules and HAB+ Ions
-0.780
v,eV
--K eV
9.514 9.335 8.111
3.996 3.678 5.182
3.899 3.271 3.289
343.921 72 343.781 00 343.779 72
6.607 5.840 5.152
4.326 4.273 4.825
3.108 3.905 3.696
HCO+ HOC+ TS
112.967 68 112.920 64 112.834 40
11.021 9.544 8.301
15.983 14.128 16.282
3.328 2.528 2.846
HOW HSiO+ TS HCSf HSCf TS
364.111 30 363.983 83 363.910 73
7.095 7.378 5.440
12.253 13.382 14.356
2.065 3.188 2.861
435.611 32 435.435 5 1 435.481 03
7.506 7.059 6.354
13.271 13.515 14.113
3.674 3.058 3.012
HBO HOB TS
100.166 57 100.091 40 100.021 74
9.344 7.075 6.315
4.673 2.242 4.462
3.290 2.163 2.875
HAlO HOAl TS
317.299 82 317.388 36 317.209 01
6.229 5.170 4.428
4.549 3.266 5.016
2.867 1.733 2.642
HNN+
TS
109.135 64 109.058 86
11.100 10.689
15.853 16.460
3.409 3.016
HBS HSB TS
422.791 71 422.637 48 422.641 79
6.799 6.392 4.780
4.119 4.121 4.464
3.301 2.186 2.759
HAlS HSAl
640.013 21 639.996 27 639.916 22
5.309 5.225 3.61 1
4.122 4.116 4.846
3.045 1.797 2.702
114.725 57 114.577 45 114.526 00
8.218 5.199 4.320
4.777 2.661 3.077
2.001 0.858 1.605
system HCN HNC TS
-E,au 92.876 68 92.859 16 92.796 11
HNSi HSiN TS
TS HBeF HFBe
TS
"
' I '
"
' ' " '
I
s
z
1 .o
0.50
0.0 0.00
45.0
90.0
135
135
90.0
45.0
0.00
180
180
HCN angle (deg)
HSiN angle, deg
Figure 2. Variation of electronic energy (Eel) and nuclear repulsion energy (VJ of HCN with B(HCN).
Figure 4. Variation of electronic energy (Eel) and nuclear repulsion energy (VnJof HSiN with B(HSiN). -1.0
1
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
;0.25
6.5
0.2
6.0
0.15
2
5.5
0.1
n
4
g'
5.0
4.5
0.00 HNSi
45.0
90.0 HSiN angle, deg
135
180
0.0 0.00
HSiN
Figure 3. Dependence of total energy (E, left scale), hardness and chemical potential 01)of HSiN upon B(HSiN).
(v),
their own respective extrema. It was further shown by Gazquez et a1.6 that, under conditions of constant chemical potential, the hardness will be maximized where Eel is a minimum and the opposite will be true when the electronic energy is a maximum. The same conclusion is derived for the case where p is not constant, but it has a maximum or a minimum at the same point where the electronic energy does. Figure 2 illustrates the behavior of these two components of the total energy, again as a function of the B(HCN) angle. It is immediately apparent that Eel and Vnn are nearly perfect mirrors of one another, with one having a maximum at almost the same point as the minimum occurs in the other. Perhaps of greater importance, the extrema in these two quantities occur at 8 = 31", 55", and 99", the same locations as the extrema in 7 and p. The maxima in 7 and Vnncoincide as do those in the p,Eel pair. We turn now to case ii with the low barrier, one example of which is the HSiN-HNSi isomerization. As illustrated by the energy profile in Figure 3, the barrier separating the 0" and 180" minima is very small, less than 1 kcal/mol.21 Nonetheless, just as in the HCN case, the hardness profile is maximized at the two energy minima and passes through two minima, separated by a maximum that occurs for 8 between 50" and 55". The transition state geometry at 143" is nearly 30" removed from the closer of these two minima. The behavior of the chemical
HNC
-0.05
45.0
90.0 HCN angle (deg)
135
180 HCN
Figure 5. Variation of two-center (ZAB)and three-center ( Z A ~ ~bond ) indices of HCN with B(HCN).
potential once again nearly mirrors that of 7. Also similar to the HCN case is the behavior of the electronic and nuclear energies of HSiN in Figure 4. While the extrema in the 7 and p profiles do not coincide exactly to the Vnn and Eel profiles, there remains a strong similarity in shape. There is also a new feature that appears in the chemical potential of HSiN: p is nearly constant over the range 53-67", coupled with a gradual change in 7 . The variation of the bond indices of the bond being broken (H-N) and formed (H C) along the reaction coordinate of HCN is exhibited in Figure 5 . As expected, IHNdecreases and IHCrises as the hydrogen moves from N to C. The three-center bond index IHCNreaches its maximum at the intersection of the two-center bond index profiles (0 = 60"). This confluence probably arises because it is at this point that the electrons are most delocalized over the three atoms. Similar results have been found for the HSiN-HNSi isomerization reaction as well. The VMprofiles (not shown) pass through a minimum; however, its position is far from the position of the minimum in the energy profile. In conclusion, the MHP is found to be valid for the isomerization reactions studied here, with a number of exceptions. The chemical potential is a poorer indicator of the preferred isomer or of the presence of the transition state. The
8124 J. Phys. Chem., Vol. 99, No. 20, 1995 extrema in the hardness profile or the chemical potential are more tightly connected to the electronic and nuclear repulsion energies than to their sum.
Acknowledgment. We are grateful to the National Institutes of Health (GM29391) for financial support. References and Notes (1) Pearson, R. G. J . Chem. Educ. 1987, 64, 561. (2) Parr, R. G . ; Chattaraj, P. K. J . Am. Chem. Soc. 1991, 113, 1854. (3) Dutta, D. J . Phys. Chem. 1992, 96, 2409. (4) Pearson, R. G.; Palke, W. E. J . Phys. Chem. 1992, 96, 3283. (5) Parr, R. G.; Gazquez, J. L. J. Phys. Chem. 1993, 97, 3939. (6) Gazquez, J. L.; Martinez, A,; Mendez, F. J . Phys. Chem. 1993, 97, 4059. (7) Pal, S.; Vaval, N.; Roy, R. J. Phys. Chem. 1993, 97, 4404. (8) Pearson, R. G. Ace. Chem. Res. 1993, 26, 250. (9) Galvan, M.; Pino, A. D., Jr.; Joannopoulos, J. D. Phys. Rev. Lett. 1993, 70, 21. (10) Nath, S.; Sannigrahi, A. B.; Chattaraj, P. K. J . Mol. Struct. (THEOCHEM) 1994, 306, 87. (1 1) Chattaraj, P. K.; Nath, S.; Sannigrahi, A. B. Chem. Phys. Lett. 1993, 212, 223.
Kar and Scheiner (12) Nath, S.; Sannigrahi, A. B.; Chattaraj, P. K. J . Mol. Struct. (THEOCHEM) 1994, 115, 65. (13) Chattaraj, P. K.; Nath, S.; Sannigrahi, A. B. J . Phys. Chem. 1994, 98, 9143. (14) Hong, H. S.; Cave, R. J. J . Phys. Chem. 1994, 98, 10036. (15) Musaev, D. G.; Yakobson, V. V.; Charkin, 0. P. Zh. Strukr. Khim. 1991, 32, 3. (16) Par, R. G.; Donnelly, R. A,; Levy, M.; Palke, W. E. J. Chem. Phys. 1978, 68, 3801. (17) Pearson, R. G. Proc. Narl. Acad. Sci. U.S.A. 1985, 82, 6723. (18) Gonzalez, C.; Schlegel, H. B. J . Phys. Chem. 1990, 94, 5523. (19) (a) Sannigrahi, A. B. Adv. Quantum Chem. 1992, 23, 307. (b) Sannigrahi, A. B.; Kar, T. Chem. Phys. Lett. 1990, 173, 569. (c) Kar, T. J . Mol. Srruct. (THEOCHEM) 1993. 283, 313. (20) Sannigrahi, A. B.; Nandi, P. K. Chem. Phys. Lerr. 1992,188, 575. (21) Inclusion of electron correlation changes this value significantly: F(MRD-CI) = 13 kcaYm01.l~ As mentioned earlier, however, the motivation of the present investigation is not the accuracy of the energetics but rather identification of common trends in the energy and hardness profiles. JP943385E
