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J. Phys. Chem. 1992, 96, 3283-3285
Support for a Principle of Maximum Hardness Ralph G.Pearson* and William E.Pake Chemistry Department, University of California, Santa Barbara, California 93106 (Received: October 29, 1991)
Computations at the Hartree-Fock level on ammonia and ethane verify the principle of maximum hardness as developed by Parr and Chattaraj. The point groups of the molecules are determined by maximum hardness. However, the equilibrium bond angles and distances are determined by the electrostatic Hellman-Feynman theorem.
Introduction Recently, Parr and Chattaraj have given a rigorous proof for a principle of maximum hardness in a chemical system (collection of nuclei and electrons).l The hardness, 7, and the electronic chemical potential, p, are defined by
TABLE I: Asymmetric Distortions of Ammonia"
-0.10b -0.02
equil' +0.02 +0.10
where N is the number of electrons and u is the potential due to the nuclei, plus any external p o t e ~ ~ t i a l . ~ . ~ The proof states that a system at a given temperature will evolve to a configuration of maximum 7,provided v and p remain constant. There is a great deal of qualitative e v i d e n ~ eand ~ , ~some semiquantitative evidence6supporting such a conclusion. Indeed, the qualitative evidence suggests that a considerable relaxation of the constraints is permissible. However, there are no accurate calculations testing the principle, especially with regard to the importance of the constraints. This paper is an attempt to provide such a test, using Hartree-Fock molecular orbital calculations, and the approximationsS
HOMO
CLUMO
- 2 ~
21
0.4164 0.4160 0.4160 0.4160 0.4164
0.2924 0.2988 0.2991 0.2988 0.2924
0.1240 0.1172 0.1169 0.1172 0.1240
0.7088 0.7148 0.7151 0.7148 0.7088
0.4136 0.4156 0.4160 0.4156 0.4136
0.2950 0.2984 0.2991 0.2984 0.2950
0.1186 0.1172 0.1169 0.1172 0.1186
0.7086 0.7140 0.7151 0.7140 0.7086
N-H bond distance
HNH bond angle -8.3d -3.3
equil' +3.3 +8.3
"All values are in atomic units of energy (hartrees). bThe distortion, in bohrs, of the N-H bond length. 'Experimental values. dThe angle given is the change in (HINH2and (H,NH, (in degrees). TABLE II: Svmmetric Distortions of Ammonia" ~
-CHOMO
CLUMO
- 2 ~
21
0.4421 0.4298 0.4164 0.4160 0.4156 0.4144 0.4090 0.4020
0.3491 0.3403 0.3022 0.2991 0.2959 0.2844 0.1811 0.0851
0.0930 0.0895 0.1142 0.1169 0.1197 0.1300 0.2279 0.3169
0.7912 0.7701 0.7186 0.7151 0.7115 0.6988 0.5901 0.4871
0.4794 0.4160 0.3907 0.3795 0.3683
0.2887 0.2991 0.2962 0.2936 0.2901
0.1907 0.1169 0.0945 0.0859 0.0782
0.7681 0.7151 0.6869 0.6731 0.6584
N-H bond distance
The C'S refer to the orbital energies of the highest occupied and the lowest unoccupied orbitals. Only filled subshell cases are considered. The necessity to use approximations comes from the derivation of (1) from density functional theory. The exact calculation of 7 and p, using (l), is not straightforward. Two molecules, ammonia and ethane, were selected for testing. The general procedure was to start with the molecules in their equilibrium geometry and to make a small displacement from equilibrium along directions given by vibrational symmetry coordinates. The orbitals, and their energies, were calculated for this new geometry, and eq 2 was used to calculate 7 and p. The changes from equilibrium values qe and pe were then noted. By picking the complete set of symmetry coordinates, we tested all possible changes in the equilibrium geometry.
1.30b 1.50 1.89 1.91' 1.93 2.00 2.50 3.00
HNH bond angle 87.4 106.7' 113.9 117.0 120.0
'All values in atomic units of energy. bThe N-H distance in bohrs. 'The equilibrium value.
Details of the Calculations Hartree-Fock wave functions were obtained using Slater type basis functions with fully optimized orbital exponents. Computations were camed out using programs written by Stevens.' The bulk of the calculations have been described p r e v o ~ s l y . ~The ~~
computations on ethane employed a minimum basis set. The ammonia study used a double-zeta plus d orbital basis. For each molecule, computations were performed at the equilibrium geometry and at molecular geometries distorted along vibrational symmetry coordinates. The distortion step sizes were chosen to obtain reliable derivatives of the energy components of the molecules.
(1) Parr, R. G.; Chattaraj, P. K. J . Am. Chem. SOC.1991, 131, 1854. (2) Parr, R.G.; Donnelly, R. A.; Levy, M.;Palke, W. E. J . Chem. Phys. 1978.68, 3801. (3) Parr, R. G.; Pearson, R. G . J. Am. Chem. SOC.1983, 105, 7512. (4) Bartell, L. S.J . Chem. Educ. 1968,45, 754. Pearson, R.G. J . Chem. Educ. 1987,64, 561. Burdett, J. K.;Coddens, B. K.; Kulkarni, G . V. Inorg. Chem. 1988, 27, 3259. ( 5 ) Pearson, R. G. Proc. Natl. Acad. Sci. U S A . 1985,82, 6723. ( 6 ) Zhou. Z.; Parr, R.G. J. Am. Chem. Soc. 1989, I l l , 7371; 1990,112, 5270. (7) Stevens, R. M.J. Chem. Phys. 1970, 52, 1397. (8) Kirtman, B.; Palke, W. E.; Ewig, C. S. J . Chem. Phys. 1976,64, 1883.
Results and Discussion Tables I and I1 show the results for ammonia. Quite different behaviors were found for the asymmetric distortions of E species and the symmetric distortions of A I species. Figure 1 shows a simplified diagram of the four normal modes of a pyramidal molecule, such as NH3, which help to understand the differences.
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(9) Chipman, D. M.;Kirtman, B.; Palke, W. E. J . Am. Chem. Soc. 1980, 102, 3377.
0 1992 American Chemical Society
Pearson and Palke
3284 The Journal of Physical Chemistry, Vol. 96, No. 8,1992 TABLE I V Symmetric Distortions of Ethane"
A1
A1
E
E
Figure 1. Normal modes of vibration of pyramidal molecules, such as ammonia. TABLE 111: Asymmetric Distortions of Ethanea eauilibrium A;, stretch, f 6 b Alu bend, i6 E, stretch, f 6 E, bend, f 6 E, rock, f 6 E, stretch, f b E, bend, *6 E, rock, f b A,, twist, f60"
HOMO
~LUMO
0.4774 0.4763 0.4732 0.4750 0.4734 0.4740 0.4633 0.4734 0.4630 0.4737
0.5856 0.5810 0.5867 0.5548 0.5776 0.5850 0.5391 0.5776 0.5780 0.5669
-2r 0.1082 0.1047 0.1135 0.0798 0.1042 0.1 1 IO 0.0756 0.1042 0.1150 0.0932
211 1.0630 1.0573 1.0599 1.0298 1.0510 1.0590 1.0024 1.0510 1.0410 1.0406
Details of these distortions a All values in atomic units of energy. are given in ref 8. CThese values are for eclipsed ethane.
The asymmetric distortions differ from the symmetric ones in two respects. First of all, positive deviation from equilibrium produces a configuration which gives the same average potential as that of negative deviation. Therefore, tHOMO, tLUMO, v-, and p must be the same for both, as seen in Table I. If we let Q represent a symmetry coordinate, then it follows that (6p/6Q) = 0 and (6q/6Q) = 0, at Q,. The equivalence of positive and negative deviations is easily seen in modes such as the E bond stretch in Figure 1. It is not as obvious in the second component of the E stretch (not shown). But it follows from symmetry that it must be so. Secondly, if we expand the energy as a power series in AQ, the linear term must vanish, and the first nonvanishing term is the quadratic one.I0 The proof for this can readily be extended to show that both (6u,,/SQ) and (6un,/6Q) are equal to zero (see Appendix). Here u,, refers to the average potential of the nuclei acting on the electrons, and u,, is the nuclear-nuclear repulsion term. Hence, for the non-totally symmetric distortions we have met both of the restrictions of Parr and Chattaraj, in that p and u,, are constant for small changes. Accordingly, the hardness, 7, should be a maximum at the equilibrium geometry. Table I shows that this is indeed the case. We see also that p is a maximum, but this need not always be true. The only requirement imposed on p is that it be constant throughout the systems2 The totally symmetric distortions give different results. Neither p nor q shows any sign of a maximum or minimum near the equilibrium geometry. The hardness keeps increasing steadily as the nuclei approach each other and would probably reach a maximum value when all the nuclei coalesced. This does not happen because at some value of Q the sum (6ue,/6Q) + (du,/bQ) is equal to zero, when averaged. Thus, the equilibrium value of Q is determined by the electrostatic Hellman-Feynman theorem and not by the maximum value of p. This is not a violation of Parr and Chattaraj's proof, since neither p nor u,, is constant. A more stringent test comes in the calculations for ethane. There are 12 different symmetry coordinates; three are symmetric and nine are asymmetric. The results are shown in Tables I11 and IV. They are essentially the same as for ammonia. All of the non-totally symmetric modes have a maximum value of the hardness at the equilibrium geometry, as predicted by the principle. As expected, 1.1 is found to be either a maximum or minimum, with respect to the various distortions. The internal rotation of A,, species is shown in Table I11 as the eclipsed form of ethane. The value of p is less than that of ~~~
(IO) Pearson, R. G. Acc. Chem. Res. 1971, 4, 152
C-C stretchb 1.30 2.10 2.70 2.90d 2.99 3.01 3.07 C-H stretchb 2.01 2.04 2.07d 2.10 2.13 HCC angle' -2.36' -1.18 equil +1.18 +2.36
-(HOMO
(LUMO
-2r
211
0.2740 0.4193 0.4665 0.4774 0.4816 0.4822 0.4763
0.3229 0.5026 0.5696 0.5856 0.5918 0.5947 0.5763
0.0489 0.0833 0.1031 0.1082 0.1102 0.1 125 0.1000
0.5969 0.9219 1.0361 1.0630 1.0734 1.0769 1.0526
0.4885 0.4829 0.4774 0.4720 0.4667
0.6143 0.5998 0.5856 0.5717 0.5581
0.1253 0.1 169 0.1082 0.0997 0.0914
1.1028 1.0827 1.0630 1.0437 1.0248
0.4794 0.4785 0.4774 0.4760 0.4744
0.5977 0.5916 0.5856 0.5796 0.5737
0.1183 0.1131 0.1082 0.1036 0.0993
1.0771 1.0701 1.0630 1.0556 1.0481
"All values in atomic units of energy. *Bond distance in bohrs. Bond angles in degrees. dEquilibrium value. e Deviation from equilibrium value.
the more stable staggered form. Also, v is a minimum at this value of the angle of rotation. Notice also in Table I1 7 is a minimum at 0 = 120° (the planar form) for ammonia. Table IV has the results for the totally symmetric distortions, in the form of carbon-carbon stretches, carbon-hydrogen stretches, and the HCC bond angle changes. The last two behave the same as do those of NH3. Neither (Sv/SQ) nor ( 6 p / 6 Q ) is equal to zero at Q,. The C-C stretch does have ( 6 p l b Q ) = 0 and a maximum value of 7 a t a bond distance of 3.01 au, somewhat greater than the equilibrium value. It is likely that this is fortuitous, and not an example of the principle as proved, since u,, certainly does not remain constant. Note that v continues to become smaller as the C-C distance decreases. In the limit of coalescence, we would form MgH6 since the C-H distance is held constant. The unstable MgH6 is expected to have a small HOMO-LUMO gap. It would appear that the principle of maximum hardness is obeyed, if the conditions of Parr and Chattaraj (constant u and p) are obeyed. It follows that the existence of symmetry in a molecule depends on the hardness. If the hardness decreases upon any distortion that destroys an element of symmetry, that element is stable. If the hardness increases, the molecule distorts and the element vanishes. While the point group of the molecule is determined by q, the equilibrium values of the bond distances and bond angles are not. They are fixed by the conditions that the repulsive force on each nucleus due to the other nuclei is just balanced by the attractive force due to the electron tloud. The principle of maximum hardness is not obeyed. These results bear a striking similarity to molecular structures deduced from the second-order Jahn-Teller There symmetry arguments can be used to predict the correct point group, but not the bond angles and distances. The guiding principle in the second-order Jahn-Teller effect is the energy gap between the ground state and the first excited state of the same multiplicity. The relationship of this gap to the hardness is self-evident. There still are reasons to believe that increasing hardness accompanies the approach of a chemical system to its equilibrium state. To illustrate this, consider the process (3) N(g) + 3 H ( d = NH,(g) which is very favorable, energetically. The overall changes in p ( 1 1 ) Pearson, R. G. Symmerry Rules for Chemicul Reactions; Wiley-Interscience: New York. 1976; Chapters 1 and 3.
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J. Phys. Chem. 1992, 96, 3285-3293 and q for reaction 3 can be found from the approximation2s3 p = (I A)/2; q = (I- A)/2 (4) where I and A are the experimental, vertical ionization potentials and electron affinities. In atomic units of energy, we find that p changes from -0.264 to -0.096 au upon going from the atoms to the molecule. At the same time, q increases from 0.236 to 0.301 au.I2 The data in Table I are not quite compatible with the experimental data. The theoretical value for p, is -0.060 au, and for q, it is 0.360 au. The discrepancies are primarily due to the failure of Koopman's theorem, applied to the electron affinity of NH3. Using either set of data, it is clear that the movement of the system toward the more stable configuration is accompanied by an increase in the hardness. This is not an isolated example. Examination of a large number of examples where a few atoms
+
(12) In a system which is a mixture of atoms or molecules, the smallest value of I and the largest positive value of A are taken. Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L. Phys. Rev.Letf. 1982, 49, 1691.
form a simple molecule always seems to show the same behavior.13 Even though p and u,, are far from constant, the hardness is increasing toward a maximum value. The equilibrium condition, however, need not correspond to an exact maximum.
Appendix After a small displacement, AQ, from the equilibrium geometry, the energy of the system may be written as E = Eo
+ (+olsU/sQl+o)AQ...
(5)
Since the potential energy, U,is totally symmetry, W/aQ must have the same symmetry as Q. Since the wave function, +o, is nondegenerate, its square is totally symmetric. Therefore, if Q is non-totally symmetric, the average value in ( 5 ) will equal zero upon integration. This applies separately to each component of u which varies with Q, that is, to u,, and unn. (13) This is not true for the formation of very large molecules or crystals of a solid.
Ro-vibronic States in the Electronic Ground State of CO2+(X2D,) Gilberte Chambaud,? Wolfgang Gabriel, Pave1 Rosmus,* Fachbereich Chemie, Johann Wolfgang Goethe Uniuersitat, 0-6000 Frankfurt, Germany
and J&lle Rostas Laboratoire de Photophysique Molgculaire, Bat 21 3, Uniuersitg Paris-Sud, 91405 Orsay, France (Receiued: July 22, 1991; In Final Form: December 30, 1991)
From the adiabatic three-dimensional potential energy functions of the Renner-Teller components for the electronic ground state X2n, of C02+the ro-vibronic levels have been calculated by a variational approach considering full dimensionality, anharmonicity, rotation-vibration, electronic angular momenta, and electron spin coupling effects. The calculated revibronic levels for J = P agree to within about 10 cm-l with the experimental data for energies up to 4300 cm-' and the set of these term values up to W = 9 is given. The electron-nuclear motion and anharmonicity coupling effects have been investigated by an analysis of the composition of the ro-vibronic variational eigenfunctions. Due to the anharmonic couplings the vibronic levels can be grouped in blocks of Fermi polyads. This finding considerably facilitates the assignments of the highly excited ro-vibronic states in the electronic ground state of COz+.
I. Introduction The coupling effects in the degenerate open shell electronic states of linear triatomic molecules have been mostly studied (cf. review articles by Jungen and Merer' and by Brown and Jorgensen2 in terms of effective bending Hamiltonians. To date, the experimental information in none of such states has been sufficient to derive three-dimensional adiabatic potential energy functions (PEF) for both components of the electronically degenerate state. In such a situation the ab initio computations can provide a valuable complement to the experimental studies. Using the recently developed three-dimensional variational approach for the calculation of the ro-vibronic spin split Renner-Teller spectrum3v4 and the theoretical PEF's, the ro-vibronic states with many vibrational quanta can be directly analyzed from the composition of the complete ro-vibronic wave functions. In the present work such an approach has been applied to the ro-vibronic states of
coz+.
This ion has been studied in numerous experiments. The UV emission spectrum extending from 280 to 500 nm was discovered by Fox, Duffendak, and Barker5 and ascribed to C 0 2 +by Duffendak and Smithe6 The most extensive work was performed in 'Permanent address: LPCR, UniversitE Paris-Sud, 91405 Orsay, France.
0022-3654/92/2096-3285S03.00/0
Mulliken's laboratory during the 1940s. Bu-Sanllehi' assigned the strong doublet around 290 nm to the B2Z:-X211, transition and rotationally analyzed the 0-0 band. Mrozowski* studied the A-X system which extends from 280 to 500 nm. He listed about 130 bands, rotationally analyzed about 100 of them, and assigned most of them to the main A(ulOO)-X(ulOO) progression. Later, Johns9 revised and extended Mrozowski's analysis by identifying bands involving v3, namely A(ulO1)-X(001) (ul = 0, 1) and A(u100)-X(u,02) (u,(A) = 0, 1 , 2 , 3 and ul(X) = 0, 1,2). Bands ( I ) Jungen, Ch.; Merer, A. J. In Molecular Spectroscopy; Modern Research; Rao, K.N., Mathews, C. W., Eds.; Academic Press: New York, 1976; Vol. 11. (2) Brown, J. M.; Jsrgensen, F.;In Advances in Chemical Physics; Prigogine, I., Rice, S . A., Eds.; Wiley: New York, 1982; Vol. 52. (3) Carter, S . ; Handy, N. C. Mol. Phys. 1984, 52, 1367. (4) Carter, S.; Handy, N. C.; Ramus, P.; Chambaud. G. Mol. Phys. 1990,
-.( 5 ) Fox, G. W.; Duffendak, 0. S.; Barker, E. F.Proc. Nail. Acad. Sci.
71. 605. ---
V.S.A. 1927, 13, 302. (6) Duffendak, 0. S . ; Smith, H. L. Phys. Rev. 1929, 34, 68. (7) Bueso-Sanllehi, F. Phys. Rev. 1941, 60, 556. (8) Mrozowski, S . Phys. Rev.1941,60,730; 1942,62,270; 1947,72,682, 691. (9) Johns, J. W.C. Can. J . Phys. 1961, 39, 1738.
0 1992 American Chemical Society
