J . Phys. Chem. 1986,90, 5524-5529
5524
Geometry Optimization Uslng Local Density Functional Methodst Brett I. Dunlap Code 6129, Naval Research Laboratory, Washington, D.C. 20375-5000 (Received: February 1 1 , 1986)
Recently, LCAO local density functional (LDF) methods have been shown to yield ground-state geometries in good agreement with experiment for a number of different molecules. However, in order to obtain accurate potential surfaces over the range of geometries that will be necessary for studies of problems such as photodissociation and relative conformational energies, the errors in current LDF techniques must be reduced and quantified. Several of the novel questions that arise in seeking to produce improved LDF techniques are discussed. Particular attention is given the important practical problem of the nonanalyticity of current LDF's, and to the numerical sampling techniques that are used to deal with this problem.
1. Introduction The linear combination of Gaussian-type orbitals (LCGTO) X a method has been shown to produce accurate values of the equilibrium bond distance, dissociation energy, and harmonic force constant in a wide variety of diatomic molecules.'-" Recently this approach has also been successfully applied to the electronic structure of a few polyatomic molecules and clusters in the vicinity of their ground-state minima,I2-l6 and to selected excited-state electronic proper tie^;'^-'^ related methods have been developed for the treatment of extended monolayers20v2'and polymer^.^^,^^ LCGTO local density-functional methods have an approximate N3 dependence of the computational time on the number of basis functions N used to describe the system, as compared to the N4 dependence of HartreeFock (HF). The smaller power law gives the density-functional approach an advantage in speed over conventional ab initio methods, which becomes greater with increasing size of the molecular system. This makes the treatment of large clusters and molecules possible in the method. Such larger systems include PdlOHz4(using model potentials), iron pentacarbonyl and nickel t e t r a c a r b ~ n y l and , ~ ~ ferrocene.26 A comparison with experiment and HF for bond distances and symmetric-stretch frequencies of the latter two molecules is given in Table I. For Ni(C0)4, wIis a nearly pure C-O stretch frequency and o2is a nearly pure Ni-C stretch frequency. The orbital basis sets used in these X a (a = 0.7) calculations are quite large and minimally c ~ n t r a c t e d .These ~ ~ results together with the earlier work on smaller molecules show that Xa and other local density functional (LDF) methods3I warrant serious attention by the chemical community at large. However, there are significant fundamental problems remaining with L D F methods, which use density functional expressions derived from solutions of the electron gas problem32for the exchange and correlation energy. The multiplet problem, which leads to ambiguity in spectroscopically labeling degenerate density functional solutions, is reviewed elsewhere33and is only a practical problem in treating chemical reactions with multiradical character. A more ubiquitous problem is the nonanalyticity of the density functional expressions for exchange and correlation. In X a 3 4 the expression for exchange and correlation is
E,, = -%or(3/8?r)1/3Sp413(r) dr with a set to 0.7 herein. Because of the presence of the 4/3 power of the density p in eq 1, current computational methods treat this term numerically. Numerical treatment requires picking out a set of sampling points in the three-dimensional (3D) volume of the molecule. The accuracy of the best current LDF methods is determined by the accuracy of this numerical integration. Greater accuracy for any given number of points can be achieved by noting that the potential that derives from the energy of eq 1 is exceedingly smooth, being proportional to the power of Presented at a symposium on the Applications of Density Functional Theory to Chemistry, Southeast/Southwest Regional Meeting of the American Chemical Society, Oct 1985, Memphis, TN.
TABLE I: Nickel Carbonyl and Ferrocene Geometries and SpectroscopicConstants Xa Hartree-Fock expt d,i
