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The first order cumulant expansion of Niemeijer and van Leeuwen is combined with a variational method to find the 'best' value of a fixed point Hamiltonian depending upon a parameter, p . A new method for calculating eigenvalues near this fixed point is proposed, based upon the neglect of terms proportional to the error neglected at the fixed point. This approach determines ap/aK unambiguously. Calculations are carried out for a few simple models. The results are more accurate than those ofdoi:10.1088/0305-4470/12/1/021 fatcat:qji5kzxq3fcn5nm7p6ltrsacga