Towards an understanding of the form of correlated wavefunctions for atoms

The transcorrelated method enables energies and properties to be found from correlated wavefunctions CΦ. The purpose here was to give some rules, for small atoms, to write down correlation functions C=Πi>jf(rij) which when multiplied into Φ, a suitable linear combination of Slater determinants, give an accurate correlated wavefunction CΦ. For the He isoelectronic series a form can be found for f(rij) in which the only correlation parameter is Z, the nuclear charge, such that CΦSCF gives transcorrelated energies within 0.003 a.u. of the exact energy (Z = 1,2,3,10). This form of f(rij) is extended to atoms with two shells, so that f depends upon parameters Z1 if ri and rj are in the first shell, and Z2 if they are in the second shell. Through independent assessments of the accuracy of these wavefunctions for Be, C, O and Ne, rules can be given for Z1 and Z2. Using these rules, the predicted correlation energies of eight such atoms were calculated and found to be reasonably accurate.