J. Phys. Chem. 1994, 98, 9143-9145
9143
Hardness, Chemical Potential, and Valency Profiles of Molecules under Internal Rotations? P. K. Chattaraj, S. Nath, and A. B. SaMigrahi' Department of Chemistry, Indian Institute of Technology, Kharagpur 721302, India Received: February 2, 1994; In Final Form: June 30, 1994"
The hardness ( q ) , chemical potential ( p ) , and molecular valency ( V M )profiles of CzH4, H3X-YH3 (X, Y = C, Si), B2H6, and H C P have been calculated ab initio using the 6-31G** basis set. These profiles correspond to rotation around the C=C bond in CzH4 and the X-Y bond in H3X-YH3, of the BHBH plane in B2H6, and of the H atom around the midpoint of the C P bond in HCP. The maximum hardness principle has been found to be obeyed in all cases. The q and p profiles are generally opposite in nature. With the exception of BzH6, the VMis maximum a t the ground state equilibrium configuration of a molecule in all cases.
Introduction The maximum hardness principle (MHP)' proposed by Pearson has received considerable attention over the past few years. It states that "there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible". A formal proof of MHP under certain constraints was given by Parr and Chattaraj.2 In a subsequent proof, Parr and GBzquez3 showed that hardness (1)will be an extremum at a point where both electronic energy (E,,) and nuclear repulsion energy ( V,,,) of a molecule reach their respective extremum values. It was further shown by GBzquez et aL4that under constant chemical potential ( p ) a system evolves to a state of maximum hardness, and 7 will also be an extremum if p and E,, are both extrema. The MHP was numerically examined by a number of investigators. Using their results of ab initio SCF calculations on NH3 and C2H6, Pearson and Palke5 showed that for small deformation in the asymmetric modes of vibration, p, electronnuclear attraction potential, u,,, and nuclear-nuclear repulsion term, u,,,, remain practically constant, and q is maximum at the equilibrium. However, for symmetric vibrations neither p nor u (u,, and u,,) is constant and 1 changes monotonically. The Pearson-Palke type observation was confirmed by Pal et a1.6 at the coupled cluster level of theory using NHs and HzO as test cases. The validity of MHP and the maximum molecular valency principle (MMVP)' was examined at the ab initio SCF level by Chattaraj et a1.8 There is no formal statement of MMVP. On the basis of earlier Chattaraj et aL8 postulated that the molecular valency will be maximum at the equilibrium geometry of a molecule. They observed that the former is valid under a variety of bonding situations, while the latter breaks down in the case of highly ionic molecules and in those cases where the inert pair effect is operative. They also accounted for the relative stability of several pairs of carbonyl-isocarbonyl by means of q values. In a recent ab initio SCF studyg Nath et al. examined the hardness and bond index profiles of the dissociation reaction H3NHF H3N + HF, and the proton transfer reaction (FH--C1)(F--H-C1)-, which are characterized by singleminimum and double-minimum potentials, respectively. Both the reactions were found to obey MHP, and the proton transfer reaction passes thorugh a transition state (TS) wherein 7 is minimum. Numerical calculations1° at the MNDO level and several other ab initio calculationsllJ2 also lend support to the MHP. A dynamical variant of MHP associated with an ionatom collision process has recently been observed by Chattaraj and Nath.l3 Ab initio calculations have shownl4 that for hardhard interactions the hard-soft-acid-base (HSAB) principle15 is a consequence of MHP.
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to Professor C. N. R. Rao on his 60th birthday. Abstract published in Advance ACS Absrracrs. August 1, 1994.
7 Dedicated @
0022-3654I94 12098-9143S04.50,/O I
,
The salient features of MHP have recently been reviewed by Pearson.16 His remark that the HOMO-LUMO gap may decrease with the extension of basis sets is corroborated by actual calculation^.^^ All these studies indicate that the MHP is potentially a very powerful tool for the study of molecular electronic structure. Although its validity has been established under a variety of molecular deformations, no work seems to have been reported on the variation of 7 under internal rotation. In the present investigation we have calculated at the ab initio SCF level the hardness profiles of C2H4 under rotation around the C=Cdouble bond, of H3X-YH3 (X, Y = C, Si) under rotation around the X-Y single bond, of HCP under rotation of H around the midpoint of the CP bond leading to the formation of HPC, and of B2H6 under the rotation of the BHBH plane leading to the planar form. Since none of the systems considered here are highly ionic, we have also investigated the nature of the molecular valency ( VM) profiles. As the variations in q are related to that in p , the chemical potential profiles of the systems have also been included in the present study.
Method of Calculation All the molecules treated here possess a singlet ground state, and consequently closed-shell SCF calculations have been performed to obtain their wave functions and to calculate various properties. The hardness and chemical potential have been calculated using the following expressions.I*Jg
where 'HOMO and ~ L U M Oare energies of highest occupied and lowest unoccupied molecular orbitals, respectively. The molecular valency is calculated from the relation7
A
B
where VAis the valency of atom A, IAB is the bond index of the AB bond, P = 2CC (C is the coefficient matrix of the occupied MO of a closed shell system and Cis its transpose), and S is the A 0 overlap matrix. In the case of C2H4, H3X-YH3, and B2Hs molecules the 6-3 lG* optimized geometries20fortheir respective ground states have been used as the starting point, and no geometrical parameters have been optimized during rotation. However, for the HCP molecule the bond angle and bond lengths have been fully optimized over the entire potential surface. During 0 1994 American Chemical Society
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The Journal of Physical Chemistry, Vol. 98, No. 37, 1994
Chattaraj et al.
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Figure 1. Variation of relative energy (AE,kcal/mol), hardness (v,eV), chemical potential (b, eV), and molecular valency (VM)of C2H4 with the torsional angle corresponding to the rotation around the C-C bond. The most stable structure (021) has been taken as the referencefor calculating the rotational barrier. rotation the spin multiplicity of the molecules has been kept invariant. All calculations have been carried out using the 6-31G** basis set.
Results and Discussion The energy (PE), p , p , and VMprofiles of the molecules under consideration are depicted in Figures 1-4. We shall first discuss the nature of these profiles separately for each molecule or each class of molecules and then attempt to generalize the findings of the present investigation. C&. The rotation of planar ethylene (D2h) around the C=C bond by 90' leads to the perpendicular form (Dz~).Figure 1 shows the corresponding AE, p , 1.1, and VM profiles. The perpendicular form lies about 113 kcal mol-' above the D2h form. It may be noted that the corresponding experimental value21 is 65 kcal mol-'. Earlier c a l ~ u l a t i o n s ~on 2 - ~conjugated ~ molecules revealed that HF/STO-3G, AM1, and M I N D 0 / 3 level theories are not adequate for the calculation of rotational barriers. Ab initio~alculations~~~2~ a t the MP2/6-3 1lG**//HF/6-31G* level produce better rotational barriers for substituted ethylenes which is, however, not true for the corresponding substituted benzenes. The discrepancy between calculated and experimental values has been arguedz6 to be due to the loss of the partial double bond character on roation in nitroethylene and nitrobenzene. In other cases the experimental values are s~spected.2~For the present problem we are not interested in the actual values of the rotational barrier heights; rather, we would like to follow the extrema1 characteristics which we believe will not alter in higher level of theories and/or by using an extended basis set. It appears that the conclusion that the energy is going to a maximum at 90° and the hardness is going to a minimum is not going to change with the level of sophistication in the calculation. The hardness decreases as one passes from the D2h to the D z form. ~ Since I.( changes only marginally, it may as well be considered as constant. The VMdecreases due to the partial loss of the double bond character of the C C bond in ethylene. HJX-YH3 (X, Y = C, Si). The internal rotation around the X-Y bond in these systems is accompanied by very small changes in AE, p , p , and V M . The corresponding profiles are shown in Figure 2. In all three cases the staggered form (D34) which represents the most stable state is characterized by maximum p and VMvalues. The p profiles of C2H6 and H3SiCH3 exhibit a minimum while a maximum is observed in the case of The small decrease in VMvalues of these molecules in going from the staggered to the eclipsed form (D3h) is due to the decrease in bond indices corresponding to the nonbonded interactions between the H atoms.
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, , ,
'.
+2,
7.26 120 iw 80 60 LO 20
0
Angle (degree)
Angle (degree)
Figure 2. Variation of relative energy (AE,kcal/mol), hardness (7, eV), chemical potential (b, eV), and molecular valency (VM)of C2H6 (-), H3CSiH3 (- - -), and Si& (- -) with the torsional angle corresponding to the rotation around the X-Y bond. The most stable structure (034) has been taken as the reference for calculating the rotational barrier.
655
-
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Angle (degree)
Figure 3. Variation of relative energy (AE,kcal/mol), hardness (7, eV), ) B2H6 with the chemical potential (1,eV), and molecular valency ( V M of rotation of the BHBH plane. The most stable structure ( 0 2 1 ) has been taken as the reference for calculating the rotational barrier. B 2 h . It is well-known that the most stable structure of diborane corresponds to the form where the BHBH plane is perpendicular to two coplanar H B H moieties. The PE, p , p, and VMprofiles of B2H6 resulting from the rotation of the BHBH plane are displayed in Figure 3. As can be seen from the figure, the planar form lies about 175 kcal mol-' above the perpendicular form, which is associated with a maximum value of p and a minimum value of p . The VMprofile shows that the molecular valency in the less stable planar form is greater than the corresponding value in the equilibrium (perpendicular) configuration. This may be due to the fact that even though the threecenter BHB bonding is considerably weakened (ZBHB(p1anar) = 0.11, IBHB(perP) = 0.28) the bond indices of the BB bond and the BH bonds in the BHBH plane increase by a greater margin in the planar form, resulting in a decrease of V M . H e - H P C . Ma et a1.28have recently carriedout very extensive calculations on the potential energy surface of HCP. They observed that the HPC isomer is a potential energy minimum at
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9145
Hardness, Chemical Potential, and Valency Profiles
in 7,p, and VMin the case of H3XYH3 molecules are only marginal and are in conformity with their very small rotational barriers. The maximum changes in 7, p, and VMoccur in the case of BzHd where the rotational barrier is the highest. In the case of C2H4 and H C P also the orderings in AE and 7 are identical. Thus, both M H P and MMVP are found to be generally valid under the internal rotations considered here.
Conclusion
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Figure 4. Variation of relative energy (A& kcal/mol), hardness ( q , eV), chemical potential ( p , eV), and molecular valency (VM)of H C P with angle HXC, X being the midpoint of the C P bond. The energy of H C P has been taken as the reference value.
TABLE 1: Calculated Values of Barrier to Rotation (AE, kcal/mol) and the Corresponding AQ (eV), A p (ev), and A V, of C2H4, HJX-YHJ (X, Y = C, Si), B ~ H Qand HCP molecule hE A7 AP A VM B2H6 C2H4 HCP C2H6 H3CSiH3 SizHs
175.5 113.5 93.4 3.2 1.5
1.o
-3.73 -3.34 -0.42 -0.04 -0.01 -0.12
2.05 0.03 -0.43 0.04 0.004 -0.13
0.19 -0.07 -0.16 -0.01 -0.01
-0.002
RMP2, RMP4, UMPZ, UMP3, and UMP4 levels, but a maximum at RMP3 and more reliable quadratic CI and Bruckner doubles levels of theory. The A.E,q , p , and VMprofiles of H C P obtained by varying the HXC (X is the midpoint of the CP bond) angle are shown in Figure 4. As can be seen, the present HF/6-31G** energy surface does not pass through a maximum (corresponding to a TS) and is thus qualitatively similar to that obtained by Ma et al.28 at the highest level of theory employed by them. Absence of a TS in the PE surface indicates that HPC is unstable with respect to bending because for a continuous curve two minima must be separated by a maximum (TS). The 7,p, and VMvalues for H C P are greater than the respective values for HPC. It is difficult to explain the presence of two minima (maxima) separated by a maximum (minimum) in the 7 ( p ) profiles. The energy varies rather slowly in the region 60° I6 5 1loo, which implies that this portion of the potential energy curve behaves somewhat like that of a TS where 7 is smaller and JpJis larger than the corresponding values of HCP. Considering the ambiguity present even in the sophisticated a b initio results28 about whether HPC corresponds to a PE maximum or a minimum, it is quitegratifying to note that the present calculation couples features of both results, viz., no TS in E profile and presence of TS in 11 profile, a consequence of MHP. Let us now examine whether there exists any correlation between barrier to rotation and the changes in 7,p, and V MThe . relevant data are given in Table 1, where the molecules are arranged in decreasing order of their barrier to rotation. As can be seen, in all cases the less stable structure is associated with a lower value of 7 and VM(BzHs is an exception). The changes
We may conclude from the results of the present investigation that the M H P which has been found to be valid for small deformations in the asymmetric modes of vibrations of molecules is alsovalid for internal rotation. It may be ascertained now with some confidence that in general the most stable configuration (like equilibrium one) of a molecule is associated with the maximum hardness value, and a transition state is associated with the minimum hardness value. Therefore, the maximum hardness condition supplements the minimum-energy criterion for molecular stability.
Acknowledgment. P.K.C. thanks CSIR, New Delhi, for financial assistance. We are grateful to the referees for constructive criticisms. 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) Parr, R. G.; Gdzquez J. L. J. Phys. Chem. 1993, 97, 3939. (4) Gdzquez, J. L.; Martfnez, A.; Mbndez, F. J . Phys. Chem. 1993,97, 4059. (5) Pearson, R. G.; Palke, W. E. J . Phys. Chem. 1992, 96, 3283. (6) Pal, S.; Vaval, N.; Roy, S. J. Phys. Chem. 1993, 97, 4404. (7) (a) Jug, K.; Buss, M. J. Compur. Chem. 1985,6,507. (b) Sannigrahi, A. B. Adv. Quantum Chem. 1992, 23, 307. (8) Chattaraj, P. K.;Nath,S.;Sannigrahi,A.B. Chem. Phys. Lett. 1993, 212. 223. (9) Nath, S.; Sannigrahi, A. B.; Chattaraj, P. K. J . Mol. Struct. (THEOCHEM) 1994, 309, 65. (10) (a) Datta, D. Inorg. Chem. 1992,31, 2797. (b) Datta, D. J. Phys. Chem. 1992.96.2409. (c) Hati, S.; Datta, D. J. Ora. Chem. 1992,57,6056. (11) GalvBn, M.; Pino, A. D., Jr.; Joannopouli, J. D. Phys. Reu. Lett. 1993. 70. 21. (12) Pino, A. D., Jr.; GalvBn, M.; Arias, T. A.; Joannopulos, J. D. J. Chem. Phys. 1993,98, 1606. (13) (a) Chattaraj, P. K.; Nath, S. Chem. Phys. Lett. 1994,217,342. (b)
Chattarai, P. K.; Nath, S. h o c . Indian Acad. Sci. (Chem. Sci.), in Dress. (14) Chattaraj, P. K.; Schleyer, P. v. R. J . Am. Chem. Soc. i994;116, 1067. (15) (a) Pearson, R.G.J. Am. Chem.Soc. 1963.85.3533. (b) Chattaraj, P. K.; Lee, H.; Parr, R. G. J. Am. Chem. Soc. 1991,113, 1855. (16) Pearson, R. G. Acc. Chem. Res. 1993, 26, 250. (17) Nath, S.; Sannigrahi, A. B.; Chattaraj, P. K. J . Mol. Struct. (THEOCHEM) 1994,306, 87. (18) Chattaraj, P. K.; Parr, R. G. In Chemical Hardness, Srrucrure and Springer: ; Berlin, 1993; Vol. 80. Bonding, Sen, K. D., Mingos, D. M. P., as. (19) For a recent review, see: Chattaraj, P. K. J . Indian Chem. Soc. 1992, 69, 173. (20) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York,1986. (21) Hollas, J. M. High Resolution Molecular Spectroscopy; Butterworths: London, 1987. (22) Hehre, W. J.; Radom, L.; Pople, J. A. J . Am. Chem. SOC.1972,94, 1496. (23) Marriot, S.; Topsom, R. D. Aust. J . Chem. 1986, 39, 1157. (24) Fabian, W. M. F . J. Compur. Chem. 1988, 9, 369. (25) Szabo, K. J. J. Mol. Struct. (THEOCHEM) 1988, 181, 1. (26) Head-Gordon, M.; Pople, J. A. Chem. Phys. Lerr. 1990,173, 585. (27) Head-Gordon, M.; Pople, J. A. J. Phys. Chem. 1993, 97, 1147. (28) Ma, N. L.; Wong, S. S.;Paddon-Row, M. N.; Lee, W.-K. Chem. Phys. Lett. 1993, 213, 189.
