The passage over a barrier is the most widely used model for a chemical reaction. We reformulate quantum barrier transmission problem in terms of a recently developed theoretical framework of the Siegert pseudostates. This results in an efficient method of treating a barrier transmission including on equal footing both resonance effects and smooth background behavior. The resonance effects are typical for double-hump barriers met in various applications. The Siegert pseudostates approach allows us to efficiently specify resonance positions and widths with high precision. The residues of the S-matrix and partial widths of resonances are expressed exclusively in, terms of the Siegert pseudostate eigenvalues. A new relation between residues for different elements of the S-matrix is found. The partial widths are important physical characteristics of a resonance since they (i) govern relative probability of resonance decay with emission of a trapped particle in left or right direction,and (ii) control the height,of the resonance maximum in the barrier transmission coefficient. (c) 2005 Elsevier B.V. All rights reserved.