Towards an understanding of the form of correlated wavefunctions for atoms

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Abstract
The transcorrelated method enables energies and properties to be found from correlated wavefunctionsCΦ. The purpose here was to give some rules, for small atoms, to write down correlation functions C=Π i>j f(r ij ) 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 CΦ SCF 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 Z 1 if ri and rj are in the first shell, and Z 2 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 Z 1 and Z 2. Using these rules, the predicted correlation energies of eight such atoms were calculated and found to be reasonably accurate.