1IntroductionResonances are temporarily trapped meta-stable states—the analogue of bound vibrational states in the continuum part of the molecular spectrum. Resonances can be formed by bound-free excitations or by collisions between reactants, hence they are manifested in both unimolecular reactions and bimolecular complex-forming reactions. Unless stabilized by collisional relaxation or spontaneous infrared emission, they will decay into products on a finite time scale that is characterized by their lifetime. In essence, resonances are quantum mechanical phenomenon because they occur at discrete energies (resonance positions), but unlike bound states they have a finite width (resonance width).Though the phenomenon of resonances has long been recognized and qualitatively understood, the quantitative determination of resonances started to appear only during past two decades. Such quantitative studies are presently limited mostly to triatomic systems. Conceptually, there are two basic approaches to determining resonance energies and widthsviacomputation. The first approach formulates the resonance problem in terms of an eigenvalue/eigenvector calculation, and has many affinities to bound-state calculations. In this approach, the resonance positions and widths are determined as the real and imaginary parts of complex eigenvalues associated with the Hamiltonian under dissipative boundary conditions. The dissipative boundary conditions can be imposed in a number of ways, principal among them being the use of a complex absorbing potential (CAP),e.g.ref. 1, a complex scaled Hamiltonian,e.g.ref. 2, or an incrementally damped matrix recursion,e.g.ref. 3. The second approach is a scattering one which relies on the calculation of the scatteringSmatrix.4–6Resonance states are associated with the complex poles of theSmatrix and thus allSmatrix related quantities such as the lifetime matrix or scattering probabilities will reflect the resonance structures in their energy-dependent profiles. Analysis of such profiles can enable the determination of resonance energies as well as widths.A third quantity which is characteristic of a resonance is the product state distributions arising from its decay. The energy of a resonance is essentially determined by the potential energy surface (PES) in the inner region, while its width depends on the coupling between the inner region and the exit channel. The product state distributions reflect scattering from the resonance into product states through the transition state region of the PES, and thus contain additional clues about the intra- and inter-molecular dynamics of the system. To specify product state distributions arising from resonance decay, the resonance eigenfunctions of the dissipative Hamiltonian must be calculated and analyzed for their amplitudes in different product channels. For the case of a bimolecular complex-forming reaction, the scattering waves associated with specific incoming reactant channels must be computed and analyzed in the asymptotic region to extract information about product state distributions. These product state distributions may comprise of contributions from one or more (overlapping) resonances which are accessed at the given scattering energy, together with possible direct scattering components. In order to fully characterize the reaction dynamics, it is necessary to consider resonance energies, widths and product state distributions for as many resonances as possible.In recent years, quantum calculations based on iterative methods have become increasingly common. These methods have better scaling properties than direct methods because they do not require explicit storage of the Hamiltonian matrix, rather only the multiplication of the Hamiltonian onto a vector. When combined with a sparse representation of the Hamiltonian such as a discrete variable representation (DVR),7both memory and CPU time can be reduced dramatically. In this perspective we will focus on an overview of the application of such iterative methods, highlighting in particular two of the most promising approaches: Chebyshev methods,e.g.ref. 8–12, and Lanczos methods,e.g.ref. 13–19. Other quantum methods2,5,6,20based on direct matrix diagonalisation techniques have also continued to develop. Due to limited space, we shall only summarise such approaches briefly. We choose two molecular systems which are dynamically quite different for illustration of the type of information that can be obtained from such detailed computations: the HCO molecule,e.g.ref. 4, 21–23, and the HO2molecule,e.g.ref. 3, 6, 15–18 and 24. Not all related references have been listed here, seeref. 25 and 26for more references for HCO. The HCO molecule represents essentially a regular system, whereas HO2represents essentially a chaotic system. Most resonances for HCO are isolated and can be assigned normal mode or local mode labels. In contrast, most resonances in HO2are overlapping and defy spectroscopic assignment. Although the quantum fluctuations appear for both systems, the underlying mechanisms are quite different.Advances in this topic have been in large part due to interplay between experiments and theories. In recent years tremendous progress has been made in detecting resonances in the fully state resolved level for both unimolecular dissociation and bimolecular reactions. Excellent reviews of the experimental studies in this field have appeared,e.g.ref. 27–29, and we limit ourselves here to quantum theoretical investigations. Additionally, we refer the curious reader to several excellent related theoretical reviews on this topic,e.g.ref. 26, 30 and 31. This review is arranged as follows: in Section 2 we describe advances in quantum iterative methods. The two case studies are discussed in Section 3, emphasizing the underlying resonance mechanisms that govern the observables. A summary with outlook to possible future developments is provided in Section 4.