Some Numerical Issues in Polymer Density-Functional Theories for Tangent Hard-Sphere Chains

The two most popular polymer density-functional theories (PDFTs), one proposed by Yu and Wu (J. Chem. Phys. 117, 2368, 2002) and the other by Chapman and co-workers (J. Chem. Phys. 127, 244904, 2007), are both applicable only to model systems based on tangent hard-sphere chains. Due to the hard-sphere repulsion, however, many integrals involved in these PDFTs contain discontinuous and/or indifferentiable integrands, and cannot be accurately evaluated by the straightforward application of any quadrature or Fourier transform. Here we show how to solve this problem in both real-space and reciprocal-space calculations, using the later versions of both PDFTs (J. Chem. Phys. 118, 3835, 2003; J. Chem. Theory Comput. 8, 1393, 2012). We also use an iteration method that can converge the PDFT equations to much higher accuracy than the commonly used Picard iteration (also known as the simple mixing or relaxation method). These allow us to obtain highly accurate results (where the maximum absolute value of the residual errors is less than 10^-10) in much less iteration steps.