4404 J. Phys. Chem. 1993,97, zyxwv 44044406 Principle of Maximum Hardness: An ab-Initio Study+ Sourav Pal,' Nayana Vaval, and Ramkinkar Roy National Chemical Laboratory, Physical Chemistry Division, Poona zyxw 41 1008, India Received: November 19, 1992 In this paper we test the principle of maximum hardness through an accurate quantum chemical calculation. Computations include extensive correlation and relaxation effects for the calculation of ionization potential and electron affinity. The molecule water has been chosen as a primary test case. Introduction relaxation effects significantly affect the results. In Fock space The principle of maximum hardness for atomic and molecular systems has been of great recent interest. Hardness 71 of a system is related to the second derivative of energy with respect to the number of particles N for constant v (potential due to nuclei plus external potential) and temperature.' The slope of such a curve in this condition is the chemical potential p. The principle states that at constant v and p, the system will evolve to a state with maximum 7. As has been pointed out,2 this principle justifies a great deal of intuition, too. A rigorous proof of this has also been given by Parr and Chattaraj.3 This principle has been tested by Pearson and Palke (PP)4 as well as Datta5 recently. While Datta used MNDO calculations, PP used ab-initio self-consistent field methods to test the principle. In each of these two cases the hardness and chemical potential were calculated through the independent particle model. zyxwvuts 1) and p are rigorously defined as1 27 zyxwvutsrqp = zyxwvutsrq (s2E/sN2)qT = (sE/bN),T (1) 7 = (IP - EA)/2 p -(IP + EA)/2 (2) Using a three point fit, it can be shown that6 where IP refers to the first ionization potential and EA refers to the first electron affinity of the N-particle system. In each of the earlier two supportingcalculations,the Koopmans' approximation has been used to estimate the IP and EA. It is well-known that while these approximations work to a reasonable degree of satisfaction, they often provide even qualitatively wrong results for IP and, in particular, for EA. The failure of Koopmans' theorem is well documented in the literature of quantum chemistry.' An exact calculation of 7 and p through eq 1 is not feasible; however, a calculation of 7 and p through eq 2 is warranted for a test of the principle. We want to carry out such a test in this paper. We may point out, however, that the rigor of this test is still limited by the fact that eq 2 is an approximate definition of p and 7. Computational Metbod One of the most accurate quantum chemical methods is the one based on the coupled cluster (CC) theory.*-10 CC methods are recommended for the efficient incorporation of electron correlation and the size extensivity, which are particularly important for extended system~.~JO For quasidegenerate systems, it is required to start from a multideterminantal model space. A general multideterminantal model space can avoid the problem of the convergence arising out of the intruder states. The Fock space based multireference CC method has been used to estimate the difference energies like the IPS, EAs, and excitation energies very accurately in a direct manner.I*-I4 This dramatically improves thesevalues. In particular, it is well-known that for the EA'S the correlation and N.C.L. Communication No. 5600. MRCC methods, therestricted Hartree-Fock (RHF) determinant of an N electron ground state is taken as the core or vacuum and the problem of (N - 1) or (N + 1) electron states is reduced to a one hole/one particle problem. (N- 1) electron problems may be considered as in the 0-particle, 1-hole Fock space sector and (N + 1) electron problems are in the 1-particle and 0-hole Fock space sector. For ( N - 1) electron states, one constructs a (N - 1) electron model space consisting of a few one-hole determinants. The model space +: can be written as (3) {+I) is a set of 1-hole determinants. The exact (N - 1) electron states may be written as (4) where TI is expressed as sum of cluster amplitudes of the 0-hole, 0-particle sector as well as new amplitudes for the 1-hole sector and the curly bracket denotes normal ordering of the operators contained withinit. Normalorderingoftheansatzwasintroduced by Lindgren.I5 The cluster amplitudes of the (0,O) sector are obtained by the projection of Schrodinger's equation for the ground-state problem to the set of N-particle excited determinants +*.8 The cluster amplitudes for the (0,l) sector may be obtained by projection of the Fock space Bloch's equation of the (0,l) sector to the virtual spaceof determinants +*. The Bloch equation of the (0,l) sector may be written as (HQ - QHy)P(O.l) = 0 (HQ - QH',;p)"''o) = 0 (5) Similarly the Bloch equation of the (1,O) sector is (6) We have used an approximation where the cluster amplitudes for the (0,l) or (1,O) sector contain only singly and doubly excited parameters and the ground-state cluster amplitudes are truncated in only a two-body approximation. This model has been sufficient to provide accurate values for medium-sized systems.12J3 Initially, we carry out a CC calculationwith only doubly excited parameters for the ground state. Then we construct a transformed Hamil- tonian H a s H = e-T(O.O)HeT(O.O) and store only the one- and two- body parts (ignoring higher body components of @. This is an additional approximation used. However, it is expected that the three body and higher body parts will not change the results significantly. Using the one- and two-body parts of H in CCD approximation, we solve the projection of eq 5 to one- and two- body virtual space of the (0,l) sector, as well as similar projection of eq 6 to the ( 1,O) sector. These are systemsof nonlinear equations furnishing zyxwv fio9'), PI), fi','), and amplitudes. The effec- tive Hamiltonian for the (0,l) and (1,O) problem may be obtained as the P-space projection of eq 5 and eq 6, respectively. In general, 0022-365419312097-4404$04.00/0 0 1993 American Chemical Society