The pseudopotential for quadrupole fields up to q = 0.9080

•New averaging and Fourier transform procedures reveal the secular frequency of oscillation for 0≤q≤0.90.•A new matrix representation of exact motion in a quadrupole field shows it can be written as the product of a lowest secular frequency (pseudopotential motion) with explicit higher frequency corrections.•A pseudopotential for motion outside the stability region is described.•The pseudopotential well depth and well width (acceptance) are clearly distinguished.•A discussion of why the acceptance cannot be calculated from the secular frequency of oscillation alone is presented.