Preliminary results on the performance of a family of density functional methods

In the quest for accurate molecular energies, the vehicle most often taken for the first leg of the journey is Hartree-Fock (HF) theory.’ This method generally yields more than 99% of the true (Born-Oppenheimer) energy, but the path for recovering the remaining one percent or so, the correlation energy, is much less clear. A wide variety of techniques are available for predicting the correlation energy, but most are quite computationally expensive relative to HF. In the LCAO approach, the cost of HF formally increases as 0( fl), where N is the number of atomic orbital basis functions. However, the formal cost of a second-order perturbation theory, or MP2,* correlation calculation is O( N’), and for more sophisticated methods such as MP4,3 QCISD(T),4 and CCSD(T),’ the cost is formally O( N’) . In practice, the cost scaling can be diminished for large systems through efficient implementation. For example, cutoff schemes reduce the cost of integral handling in HF to O(N’), and for practical computations this is the rate-limiting step. Nonetheless, the expense of many traditional correlated methods soon becomes prohibitive as N increases. Density functional theory ( DFT)6 is an alternative approach, which uses general functionals of the electron density (and possibly its derivatives) to model exchange and correlation. DFT is an exception to most correlated methods, such as the ones mentioned above, in that it provides an estimate of the correlation energy at a relatively modest cost, formally 0 ( N3>, which is reduced to 0 (N) by efficient implementation.’ Therefore, for large systems the DFT cost scales only as O(N2), as in HF (due to Coulomb integral handling). Such computational inexpense is, of course, a necessary prerequisite for correlated studies of large systems, and hence the investigation of the quality of density functional methods deserves attention. This Communication gives a preliminary report of a study comparing the performance of a family of six DFT methods with HF and the correlated ab initio methods MP2 and QCISD.4 The set of 32 molecules studied’ is a subset of that used in the validation of G2 theory,’ consisting of all the neutral systems having first-row atoms only, plus H,. Equilibrium geometries, dipole moments, harmonic vibrational frequencies, and total atomization energies were calculated for each system by each theoretical method and compared with experimental data. Complete results of these calculations and a full discussion will be given in a future publication;” here, however, we merely summarize the results for atomization energies. Two exchange and two correlation functionals are used in the present study. The simplest exchange functional is the familiar zeroth-order” Slater (S), or Xa functional.t2 We have adopted throughout the value a = 2/3, from the theory of the uniform electron gas. Attempts to improve the quality of zeroth-order functionals generally involve the introduction of gradient-correction (first-order) terms. We have chosen the first-order exchange functional of Becke (B).13 The correlation functionals are the Vosko, Wilk, and Nusair (VWN) parameterization (Recipe V) I4 of exact uniform electron gas results,” and the first-order functional of Lee, Yang, and Parr (LYP),r6 as transformed by Miehlich et al.” This leads to a total of six different DFT methods, illustrated diagrammatically in Fig. 1. The vertical axis represents the type of exchange functional (zeroth order or first order), while the horizontal axis classifies the correlation treatment (null, zeroth order, or first order). At this point we introduce a general notation for DFT methods, which is used in Fig. 1. We propose the name X-C for a method that uses exchange functional X and correlation functional C. Thus, for example, Becke exchange paired with Lee, Yang, and Parr correlation is abbreviated B-LYP. For methods which do not employ either an exchange or a correlation functional, the label “null” is used for X or C to indicate the “null functional.” Note that S-null is the method commonly referred to in the literature as HFS (Hartree-Fock-Slater), and that SVWN is also known as LSDA (local spin-density approximation). We have implemented the Kohn-Sham (KS) equations18 for each of the DFT methods in Fig. 1 in a modified version of the GAUSSIAN 92 quantum chemistry package.” The DFT densities are therefore obtained in a selfconsistent manner strictly analogous to the HF SCF procedure. Projection techniques, such as those of Dunlap