Dynamic correlation

From a knowledge of the Hartree-Fock and exact non-relativistic energies of atoms, the correlation energy E_c, as defined by Lowdin, may be calculated. For atoms this correlation is defined as dynamic correlation. The separate like-spin and unlike-spin contributions, E_cσσ, E_cαβ, 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 E_c. 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 E_cσσ, and E_cαβ components. It is immediately apparent that a new functional, OPTC, derived from the P91 components as 0.6625 x E_cσσ, + 1. 1015 x E_cαβ is an improvement (its mean absolute error is only 0.006 E_h). 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 E_h. 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.     

Left-right correlation energy

We first attempt to determine a local exchange functional E_x[p] which accurately reproduces the Hartree-Fock (HF) energies of the 18 first and second row atoms. E_x[p is determined from p and |δp|, and we find that we can improve significantly upon Becke's original generalized gradient approximation functional (commonly called B88X) by allowing the coefficient of the Dirac exchange term to be optimized (it is argued that molecules do not behave like the uniform electron gas). We call this new two parameter exchange functional OPTX. We find that neither δ p or t = Σ δ _i |² improve the fit to these atomic energies. These exchange functionals include not only exchange, but also left-right correlation. It is therefore proposed that this functional provides a definition for exchange energy plus left-right correlation energy when used in Kohn-Sham (KS) calculations. We call this energy the Kohn-Sham exchange (or KSX) energy. It is shown that for nearly all molecules studied these KSX energies are lower than the corresponding HF energies, thus giving values for the non-dynamic correlation energy. At stretched geometries, the KSX energies are always lower than the HF energies, and often substantially so. Furthermore all bond lengths from the KSX calculations are longer than HF bond lengths and experimental bond lengths, which again demonstrates the inclusion of left-right correlation effects in the functional. For these reasons we prefer to split the correlation energy into two parts: left-right correlation energy and dynamic correlation energy, arguing that the usage of the words ‘non-dynamic’ or ‘static’ or ‘near-degeneracy’ is less meaningful. We recognize that this definition of KSX is not precise, because the definition of a local E_x[p] can never be precise. We also recognize that these ideas are not new, but we think that their importance has been insufficiently recognized in functional determination. When we include third row atoms in our analysis, we are unable to find a local exchange functional which is a substantial improvement over B88X for the reproduction of HF energies. This must arise from the effects of the core orbitals, and therefore we do not consider that this detracts from the improved accuracy of OPTX. We report some MCSCF calculations constructed from bonding-antibonding configurations, from which we attempt to calculate ab initio left-right correlation. There is only moderate agreement between the two approaches. Finally we combine the OPTX functional with established correlation functionals (LYP, P86, P91) to form OLYP, OP86 and OP91; OLYP is a great improvement on BLYP for both energy and structure, and OP86, OP91 are an improvement over BP86, BP91 for structure. The importance of the exchange functional for molecular structure is therefore underlined.