  The computational cost of quantum Monte Carlo (QMC) calculations of realistic
periodic systems depends strongly on the method of storing and evaluating the
many-particle wave function. Previous work [A. J. Williamson et al., Phys. Rev.
Lett. 87, 246406 (2001); D. Alf\`e and M. J. Gillan, Phys. Rev. B 70, 161101
(2004)] has demonstrated the reduction of the O(N^3) cost of evaluating the
Slater determinant with planewaves to O(N^2) using localized basis functions.
We compare four polynomial approximations as basis functions -- interpolating
Lagrange polynomials, interpolating piecewise-polynomial-form (pp-) splines,
and basis-form (B-) splines (interpolating and smoothing). All these basis
functions provide a similar speedup relative to the planewave basis. The
pp-splines have eight times the memory requirement of the other methods. To
test the accuracy of the basis functions, we apply them to the ground state
structures of Si, Al, and MgO. The polynomial approximations differ in accuracy
most strongly for MgO and smoothing B-splines most closely reproduce the
planewave value for of the variational Monte Carlo energy. Using separate
approximations for the Laplacian of the orbitals increases the accuracy
sufficiently to justify the increased memory requirement, making smoothing
B-splines, with separate approximation for the Laplacian, the preferred choice
for approximating planewave-represented orbitals in QMC calculations.
