Dynamic correlation

Abstract

From a knowledge of the Hartree-Fock and exact non-relativistic energies of atoms, the correlation energy Ec, as defined by Lowdin, may be calculated. For atoms this correlation is defined as dynamic correlation. The separate like-spin and unlike-spin contributions, Ecσσ, Ecαβ, may be calculated as a sum of pair energies from quantum chemistry; we have used the unrestricted M⊘ller-Plesset second-order algorithm, and then scaled them to give Ec. These three values may also be computed using dynamic correlation functionals, with the Stoll partitioning. The VWN, LYP and P91 functionals were studied for the atoms from H to Ar. Although the total correlation energies of LYP and P91 are similar, only P91 gives a semi-sensible breakdown into the Ecσσ, and Ecαβ components. It is immediately apparent that a new functional, OPTC, derived from the P91 components as 0.6625 x Ecσσ, + 1. 1015 x Ecαβ is an improvement (its mean absolute error is only 0.006 Eh). Using the recently introduced improved exchange functional OPTX (obtained through a fit to the HF energies of atoms), Kohn-Sham calculations were performed on the atoms using the OPT(=OPTX + OPTC) functional. The total energies have a mean absolute error of 0.006 Eh. The study then moves to molecules. First it is shown that the dynamic correlation energy contribution to the dissociation energies is very similar (within 2kcalmol−1 in most cases), whether it is calculated with LYP, P91 or OPTC. A calculation is then made of the HF contribution, the dynamic contribution through OPTC and the left-right contribution through OPTX, to molecular binding. In many cases the sum agrees with the observed value, but in some cases the prediction is significantly in error, e.g. O2 is overbound by 10 kcal mol−1. Thus either OPTX or OPTC or both are inadequate. An attempt was made to determine improved local exchange and correlation functionals by fitting to both atomic and molecular data, but this was unsuccessful. The conclusion is that the method is close to the limit of accuracy achievable from separately optimized local exchange and correlation functionals. Finally, a new hybrid functional O3LYP, which is a substantial improvement on B3LYP, is presented.