Accurate energy eigenvalues for the H2 molecule are presented here using a new approximate technique. This technique is based on a recently published two-point quasi-rational approximation method, which is an extension of the well-known Padé procedure. Approximants of order nine for the ground and excited states (n = 1, 2) have been determined. More than five exact digits are always found for n = 0 and 1, and four for n = 2. The results are much better than those of the modified shifted large-N technique and they are good even when that method fails. © 2006 Elsevier B.V. All rights reserved.