We review the formalism of matrix product states and one of its recent generalisations which allows to variationally determine the dispersion relation of elementary excitations in generic one-dimensional quantum spin chains. These elementary excitations dominate the low energy effective behaviour of the system. We discuss recent work where we show how we can also describe the effective interaction between these excitations – as mediated by the strongly correlated ground state – and how we can extract the corresponding S matrix. With these two ingredients, we can already build a highly non-trivial low-energy description of any microscopic Hamiltonian by assuming that higher order scattering processes are negligible. This allows to extract accurate information about the behaviour of the system under perturbations or at finite temperature, as we illustrate using the spin 1 Heisenberg model.