J. Phys. Chem. 1995, 99, 9337-9339
9337
Chemical Hardness in Density Functional Theory Guy Makov Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE,U.K. Received: August 30, 1994; In Final Form: November 28, 1994@
Variations of the chemical hardness and other global molecular quantities are considered in “pure-state” density functional theory. It is found from symmetry considerations that the hardness is an extremum with respect to symmetry breaking variations about a symmetric configuration of the nuclei. The relation of this result to the principle of maximum hardness is discussed.
Introduction
possible”.2 This is known as the maximum hardness principle. Parr and Chattaraj5 have proposed a proof that the hardness is Hardness’,2 is a measure of the resistance of a chemical species to changes in its electronic configuration. It is t h o ~ g h t ~ , ~ a maximum at equilibrium nuclear configuration, under conditions of constant chemical potential and temperature. This proof to be an indicator, together with electronegativity of chemical rests on the assumption that all nonequilibrium states near the reactivity and stability. It has also been claimed that the equilibrium state can be generated by small perturbations of interaction between hard species is predominantly electrostatic, the equilibrium state. Other interpretations of the maximum while between soft species it is predominantly covalent (Le., hardness principle have been considered, e.g., that the hardness through mixing of orbitals).2 An empirical measure of hardness is a maximum for the nuclear isomer with the lowest energy.6 is the energy, AE, of the disproportionation reaction: However, in this work only the formulation of Parr and Chattaraj will be considered. Numerical confirmation of the maximum hardness principle has been sought through a variety of quantum mechanics where X is any chemical species. AE is calc~lations.~-’~ All such calculations have been at constant N and at zero temperature. The only variation allowed was of &=I-A (2) the positions of the nuclei about the configuration chosen. where I and A are the ionization potential and the electron Pearson and P&e7 have suggested that the condition of constant affinity, respectively, of species X. The larger the energy of chemical potential is observed for small asymmetric vibrations the reaction, the harder the species is considered to be. It is of about a symmetric equilibrium position. Numerical calculainterest to note that reaction 1 does not proceed spontaneously tions’,*of the hardness along the asymmetric normal modes have for (Le., AE > 0) for all known chemical species.’ found it to be a maximum at the equlibrium position in Formalization of the hardness concept was carried out in the agreement with the formulation of Parr and Chattaraj. Symframework of density functional theory.’ In this theory the metric vibrations have been found to induce change in both p chemical potential, p, has been identified as the derivative of and q near the equlibrium position. More generally Pal et aL9 the energy, E, with respect to the number of electrons, N, at calculated the variation of the hardness and chemical potential constant extemal potential, v(r): in asymmetric variations of the nuclear potential about the unstable equilibrium positions of a linear water. They found p = (aE/aN), (3) that both the chemical potential and the hardness were extrema with respect to such variations. Similarly in a calculationlo of The hardness, q, has similarly been shown to be the hardness along a reaction path, it was observed that the hardness is a minimum at the transition state. The object of the present work is to consider the effect of (4) variations of the nuclear potential v(r) on the chemical potential and the hardness within the “pure-state” formulation of density In practice these definitions are often replaced with the sofunctional theory. In this framework the variables are N and called finite difference approximations which relate the chemical v(r), which correspond to the variables in most calculations. potential and the hardness to the ionization potential and the Following a brief review of several results in this formalism electron affinity: the variational equations for the quantities of interest are I obtained. The implications of symmetry on the variational /A%- /2(I+A)=-xM (5a) principles are considered in detail. The results are compared with those of Parr and Chattaraj for the maximum hardness and with the results of numerical calculations. is the Mulliken electronegativity. Note the relation of the expression for the hardness in (5b) to the empirical measure of hardness in eq 2. It has been suggested that “there seems to be a rule of nature that molecules arrange themselves so as to be as hard as
ZM
@Abstractpublished in Advance ACS Absrracts, May 15, 1995.
“Pure-State” Density Functional Theory In density functional theory the electron density, n(r), alone determines the ground state energy, EO,through the energy functional E[n]: E[n] = E,
0022-365419512099-9337$09.00/0 0 1995 American Chemical Society
(6)
Makov
9338 J. Phys. Chem., Vol. 99, No. 23, 1995 The energy functional is conventionally written out with the extemal potential, v(r), contribution written out explicitly as E[n] = F[n]
+ j v ( r ) n(r) d3r
(7)
F[n]is a universal functional of n(r). For a given v(r) there is a minimum principle with respect to variations in the electron density that conserve N . If n’ is any (suitable)’ trial density, then E[n’]
’E[n] = E,
or in the form of a variational principle 6(E[n] - p j n ( r ) d3r) = 0
(9)
p is a Lagrange multiplier associated with the constraint of
constant N , and it has been shown to be equal to the chemical potential. The Euler equation is
p = dE[n]/dn(r) = dF[n]/dn(r)
+ v(r)
(10)
If the potential is varied at constant N, we obtain
6E/dv(r) = n(r)
(1 1)
So far we have not included the nuclei-nuclei interactions in the energy. However if we do consider them then the nuclear charge density, n+(r) is
n+(r) = Czid(r - RJ
(12)
i
where zi and Ri are the nuclear charges and positions, respectively. The extemal potential is
The additional nuclear interaction energy causes the variation of E with v(r) to be
dE/dv(r) = n(r) - n+(r)
(14)
The vertical ionization potential, I , of N electrons in an extemal potential is
Z = E [ N - 11 - E [ M
(15)
and the variation of Z with v(r) is
6N6v(r) = nNp1(r)- n,,,(r)
(16)
where nM is the ground state density of M electrons in the potential v(r). The variation of the electron affinity may be similarly obtained. It is of interest to note that eq 16 is an approximate expression for the negative Fukui function associated with the reactivity of the species under considerationtoward electrophilic reagents. l 2
Variations and Extrema In the “pure-state’’ formalism, at constant N , the variational propertiesI3 of various constants, X , such as the energy, the chemical potential, hardness etc. are defined by the equation 6X 6X = j-dv(r) 6v(r)
d3r = j n ( r ) 6v(r) d3r
In general this integral does not vanish unless x(r) vanishes, in which case an extremum with respect to all variations in v(r) exists. However under certain conditions it may vanish leading to an extremum for particular types of variations. It is to the first class that the total energy at an equilibrium configuration of the nuclei belongs. The Parr and Chattaraj formulation of the principle of maximum hardness belongs to the second class, as it allows only variations of v(r) about constant p. Consider first the case of an extremum for all variations. In the case of the total energy, x(r) = n(r) - n+(r). The variational principle (17) is then well-known to be equivalent to the condition that the electrostatic forces vanish at the equilibrium positions of the nuclei. (Altematively one may say that at the nuclear positions the electrostatic repulsion of the nuclei is cancelled out by the electrostatic attraction from the electronic charge distribution.) Similarly for the chemical potential and the hardness we find from eq 16 and using the finite difference approximations (3, x(r) = (nN+l(r) - n~-1(r))/2for the chemical potential and x(r) = (nN+l(r) nN-I(r) - 2nN(r))/2 for the hardness. By analogy with the total energy case, we find that a general variational principle for the chemical potential or the hardness will exist only if the electrostatic force arising from the respective charge distributions x(r) is zero at the equilibrium positions of the nuclei. There is at present no evidence (or claim) for such behavior. Now consider the possibility of an extremum for certain classes of variations. This may arise from symmetry conside r a t i o n ~ . ‘If~ we consider a nuclear configuration which has the symmetry of the point group G, then the potential, v(r), belongs to the totally symmetric irreducible representation of G. The global quantities energy, chemical potential, hardness, etc., will be invariant under all the symmetry operations g of G. By the theorem proved in the Appendix the functions x(r) also belong to the totally irreducible representation of G (e.g., from eq 11 we find, as expected, that the ground state electron density has the same symmetry as the extemal potential). This means that if we consider any variation dv(r) which does not preserve the symmetry, Le., belongs to a different irreducible representation, then the integral in (17) vanishes and the quantity Xis at an extremum relative to this variation. In particular this implies that with respect to variations along all asymmetric vibration modes about some symmetric configuration the invariants will be at an extremum. Obviously this argument does not tell us anything about systems belonging to the C1 point group (Le., systems which have only the symmetry of the identity operation). If extrema exist for quantities other than the energy in such systems, then they arise from other principles. This last argument connects the present work with that of Parr and Chattaraj5 who found that the hardness was an extremum (maximum) when the chemical potential was constant (Le., 6p = 0). This is always the case for asymmetric variations about a symmetric configuration of the nuclei, as shown above. It might also be the case for some other variations, but these will be specific exceptions. This can be concluded from the results of numerical calculations which have shown that there exist cases where the hardness is not an extremum with respect to certain variations about the equilibrium configuration of the n ~ c l e i . ~ The - ~ symmetry-based picture is, however, more general. It predicts that all the invariants (e.g., energy, chemical potential, hardness) will be extremal with respect to asymmetric variations about a symmetric nuclear configuration. This explains the results of Pal et al.? who considered the symmetric and asymmetric variations of the nuclei about the linear configuration of a water molecule. They found that for symmetric variations neither p nor 17 are at an extremum in
+
J. Phys. Chem., Vol. 99, No. 23, 1995 9339
Chemical Hardness in Density Functional Theory this configuration. For asymmetric variations they found that both p and q were extrema, though this could change from a maximum to a minimum depending on the type of variation. Results obtained by Datta'O for the variation of the hardness along the reaction coordinate near the transition state of certain reactions are also in agreement with this picture, as both reactions involved symmetric transition states. The variation along the reaction coordinate broke this symmetry, and therefore both the chemical potential and the hardness were found to be extrema at the transition state. Finally it should be noted that symmetry principles do not determine by themselves whether such extrema will be minima or maxima. For this we have to resort to other arguments such as those of Parr and Chattaraj.
Acknowledgment. The author acknowledges useful discussions with Professor A. Nitzan and Dr. S. Crampin. A FCOClore Foundation scholarship is gratefully acknowledged, as is the hospitality and advice of Dr. M. C. Payne. Appendix Consider a function fir) which belongs to the symmetric irreducible representation of the group G, Le., for every symmetry operation g in G:I4
Iv] is a functional offir) and is invariant under the operations of the group G. To which irreducible representation does the
variation h(r) belong? Applying eq A1 we obtain
Therefore the variation h(r) also belongs to the symmetric irreducible representation. This result implies that small variations of the function f at equivalent points, result in equivalent variations in the value of the functional I , which is to be expected.
References and Notes (1) Parr, R. G.; Yang, W. Density Functional theory of Atoms and Molecules; Oxford University Press: Oxford, 1989. (2) Pearson, R. G. J. Chem. Ed. 1987, 64, 561. (3) Yang, W.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A. 1985,82,6273 (4) Pearson, R. G. Acc. Chem. Res. 1993, 26, 250. Harbola, M. K. Proc. Natl. Acad. Sci. U.S.A. 1992, 89, 1036. (5) Pan, R. G.; Chattaraj, P. K. J. Am. Chem. SOC.1991, 113, 1854. (6) Galvan, M.; Dal Pino, A.; Joannopoulos, J. D. Phys. Rev. Lett. 1993, 70, 21. (7) Pearson, R. G.; Palke, W. E J. Phys. Chem. 1992, 96, 3283. (8) Chattaraj, P. K.; Nath, S.; Sannigrahi, A. B. Chem. Phys. Lett. 1993, 212, 223. (9) Pal, S.; Vaval, N.; Roy, R. J. Phys. Chem. 1993, 97, 4404. (10) Datta, D. J. Phys. Chem. 1992, 96, 2409. (11) Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136, B864. (12) Parr, R. G.; Yang, W. J. Am. Chem. SOC.1984, 106, 4049. (13) For a review of the calculus of variations see (e.g.): Bronshtein, I. N.; Semendyayev, K. A. Handbook ofMathematics, 3rd ed.; Verlag Harri Deutsch: Frankfurt, 1985. (14) See, e.g.: Cotton, F. A. Chemical applications of Group Theory, 3rd ed.; Wiley: New York, 1990. JP942332U
